About Rome and Central Asia, Persia was always getting invaded from the north by various nomads (like the White Huns), so the Roman conquerors of Persia might have to deal with them before they can start moving into India. They decide to stamp out the nomad threat forever and focus their conquering efforts there, buying
India time to unify and driving potential Mughals into the subcontinent.
Rather unlikely that the Romans would take this course. It would distract Romans from their true strategic objective, India, plus it would stretch their supply lines greately if it occurs before they have consolidated their control of Eastern Sarmatia and the Volga line, which would occur much at the same time as encorachment into India.
Bonus points if you can have an Indian Teutoberger Wald--the overconfident Romans, fresh from conquering Persia, try to conquer India, only to have a major horse-nomad incursion into their rear that threatens their supply lines. Some Roman forces are pulled back and are squished by the nomads; other, more forward-deployed units, are squished
by Indians.
Hmm, this is already rather more plausible. But it does not need involve nomads at all, in the force that eventually unifies India. You may have a talented dynasty from some Indian state, which emerges from this confrontation, and earns enough prestige to develop a momentum as the unifiers of India against the Romans. Again, not a sure thing, and widely dependent on butterflies (much as the original teutoburg affair) but it might definitely happen.
About the Scandinavians, when would the population increases that caused the Viking Age in OTL come into effect in Scandinavia if Germany is Roman? If it comes early enough, the Romans are still busy with Persia and Sarmatia and by the time they start realizing the significance of North America, it would be too late.
Hmm, I'm not sure what the butterfly effects would be of a Roman Germania and Sarmatia were, on the time schedule of the population increase in Scandinavia that caused the Viking Age in OTL. But unless it is rather drastically accelerated, it is quite unprobable IMO that they would buy the Scandinavians the window of centuries that would allow them to colonize North America without Roman interference and build settler states strong enough to resist Roman onslaught.
Since a strong and prosperous Roman Empire would in all likelihood adopt the Norse ocean-going techniques (and they would also adopt ocean-worthy shipping from a different source, India) within a century or so from the start of the Viking expansion, and follow the Norse explorers and settlers in the Iceland-Greenland-Vinland route in realatively early "hot pursuit", so to speak.
Let's see a TL:
1st Century CE: The Roman Empire conquers Germania, Bohemia, Dacia, Nubia, Britannia-Caledonia, and Hibernia. Romanization of Northern Europe starts in earnest as the conquests spurs the discovery of various technological improvements (heavy plough, three-field system, horse collar) which allow extrensive development of Northern Europe.
2nd Century CE: The Roman Empire conquers Mesopotamia and vassalizes Persia. Western Sarmatia is conquered, and Romanization of Eastern Europe starts. Heavy settlement of Northern Europe is ongoing.
It is a time of rapid technological progress as various key discoveries are adopted from improved contact with Persia, India, and China (papermaking, blast furnace & cast iron, seed drill, hand crank, crossbow, woodblock printing) or independently developed (wheelbarrow, abacus, caliper, waterwheel & watermill, solid-treed saddle & stirrups, iron horseshoes, cranes). Renovation and expansion of the Suez canal.
3rd Century CE: A time of crisis in the Roman Empire as civil wars and plague strike the Empire. However, the expanded borders and technological improvements the Empire has achieved prevent the Crisis from causing irreversible economic and social damage that would have occurred in their absence.
Extensive reforms are applied to deal with the causes and effects of the Crisis which stabilize the Empire in the long term: a professional scholar bureaucracy is created on the Chinese model, the Vigiles are given more powers and numbers to create a counterweight to the Pretorian Gurad (and viceversa), economic reforms give tax breaks to yeoman farmers and lift restrictions to finance and commerce, strengthening the urban proprietary trading class, while taxing the landed aristocracy more heavily. The Senate's membership is broadened to give representation to landed and urban trading elites from throughout the Empire.
The Army is restructured to create a mobile force, and soldiers' pay is tied to the rate of inflation, stabilising the income of the soldiers and making them less susceptible to bribery from ambitious commanders, whilst attracting a higher standard of recruit. Veterans are guaranteed liveable land grants in the provinces when they discharge. A system of strong property rights with lease and usufruct contracts for land development akin to sharecropping is created. Reform of land ownership combines the recognition of private ownership and the rewarding of cultivators with a harvest share commensurate with their efforts.
4th Century: A revitalized Roman Empire begins a new cycle of expansion and rapid technological progress (innovations include mobile type printing, artesian wells, grindstones, horizontal loom, distillation, wine press, soap, water hammer, arched saddle, longbow, spurs): Eastern Sarmatia is conquered and Persia is annexed to the Empire for good.
An extensive canal system in Northern Europe that links the Rhine, Weser, Elbe, Oder, and Vistula and is later expanded to the Nemen, Daugava, and Dneipr. The same way, they link the Danube, Dneister, and Dneipr. Other canals link the Rhine with the Danube, the Elbe and the Oder with the Danube, and the Vistula with the Dneister. The canal system is also extended westward, too, linking the Rhine, Scheldt, Meuse, Seine, Loire, Rhone, Saone, and Garonne rivers.
Renewed confidence of the Roman people in their society results into revitalization of European polytheism: various polytheistic traditions (esp. Greco-Roman, Celtic, Germanic, Slav, and Egyptian) from throughout the Empire are merged into an universal "Romanist" system and hierarchy, which also borrows ideas from Roman philosophy, Buddhism, and Hinduism. The doctrine develops that an universal immanent divine force exists, which creates fate and natural law, and the various gods are self-aware universal archetypal expressions of natural law, wearing different faces and names in different cultures, who can partially affect fate and natural law in their field of responsiblity. Middle Eastern mystery cults (such as Christianity and Mithraism) begin to lose influence and die out. A system of unitary procedure and law, with recognised authorities to provide legal opinion and formalised educational institutions for practitioners, is developed.
5th Century: Various groups of Central Asian nomads unify in the Huns confederation and make a massive breakout in Eastern Europe. They are eventually repelled by the Roman legions, using intentional and disciplined combined arms tactics between heavy cavalry and archers. This results in gradual unsystematic Roman expansion in Central Asia. Repeated Persian uprisings are suppressed and ruthless Romanization of Persia is enforced. Persecution and suppression of Middle Eastern monotheistic religions (Christianity, Mithraism, Zoroastrism), which are seen as hostile to Roman civilization. Romanist religion spreads to become the faith of the overwhelming majority of the Roman Empire's population. Invasion and conquest of Arabia. Weirs and Dams are built on the Tigris and Euphrates allowing irrigation canals and subterranean Aqueducts to farm in the desert.
Technological progress steadily continues: buttons, mirrors, rat traps, spectacles, windmills, tidal mills, spinning wheels, magnets, compass, counterweight trebuchets, astrolabes, rib vault, coffee, hang glider, hard soap, shampoo, nitric acid, alembic, valve, reciprocating, combination lock, quilting, pointed arch, surgical catgut are developed.
Legal reforms create increasingly complex financial instruments in trade, banking and investment, including limited liability and full legal personage, and different legal systems for slavery: 'house' slaves become trusted retainers who act as commercial agents, estate administrators and other vital functionaries, perform paramilitary functions, provide skilled labour and ultimately form a stratum of 'ministerial' upper class while 'chattel' slaves remain a labour reserve or luxury consumption good (ever more costly, but ultimately disposable). A toned-down form of temporary "house slavery" is developed to provide apprenticeship: A houseborn slave (verna) shows promise in youth and is trained, either in-house or by being lent or sold to someone who has use for him (trade in gifted children is brisk). Once he has the required skills (as an accountant, merchant, administrator, physician, artisan or whatever), he works for the profit of his owner. These people only change hands rarely, and if they do it is for large sums. Traditionally, after ten to fifteen years of service (in comfortable quarters and nice conditions, with some informal pay), they are granted their freedom and continue to work for their masters, now for pay. Some may strike out on their own, though they are still bound to them by legal ties (may not compete with them or act against their interests). Many former owners will provide seed capital for their freedmen. Free-born but poor children join this system by temporary slavery contracts that provide legally-enforceable guarantees of liberation after a fixed term of service and some basic personal rights for the temporary slave.
The "Roman Agricultural Revolution" starts. Roman traders and explorers travel across most of the Old World, and establish an early global economy across most of Asia and Africa and all of Europe, with their trade networks extending from the Atlantic Ocean and Mediterranean Sea in the west to the Indian Ocean and China Sea in the east. The global economy established by Roman traders across the Old World, enable the diffusion of many crops and farming techniques among different parts of the Roman world, as well as the adaptation of crops and techniques from and to regions beyond the Roman world. Crops from Africa such as sorghum, crops from China such as citrus fruits, and numerous crops from India such as mangos, rice, cotton and sugar cane, are distributed throughout Roman lands, which previously had not grown these crops. Romans start developing a scientific approach to agriculture based on three major elements: sophisticated systems of crop rotation, where land is cropped four or more times in a two-year period, highly developed irrigation techniques, using machines such as norias, water mills, water raising machines, dams and reservoirs, which allow to greatly expand the exploitable land area, and the introduction of a large variety of crops which are studied and catalogued according to the season, type of land and amount of water they require. Manufacture of silk spreads in the Roman Empire.
6th Century: Continued conquests of territory into Central Asia. Persia subsides into an uneasy peace. Expansion through Persian territory into India. Lenghty wars to secure it. Merging of Romanist and Hinduist religions. The Kingdom of Aksum is invaded and conquered by Rome. Plague hits again the Empire, delaying expansion in Asia but its effects are diminished by the effectiveness of the Roman public health system and Roman culture's focus on personal hygiene. The plague spurs interest into medicine and natural philosophy in Roman culture.
Technological progress continues, with the development of the hourglass, mechanical clocks, flywheel, crankshaft, connecting rod, and water turbine, many numerous innovative industrial uses of water mills, early industrial uses of tidal power, wind power, and fossil fuels such as petroleum, and the earliest large factory complexes. A variety of industrial mills are invented in the Roman world, including fulling mills, gristmills, hullers, paper mills, sawmills, shipmills, stamp mills, steel mills, sugar mills, tide mills, and windmills. Roman engineers also invent crankshafts and water turbines, first employ gears in mills and water-raising machines, and pioneer the use of dams as a source of water power, used to provide additional power to watermills and water-raising machines.
Many industries are generated due to the Roman Agricultural Revolution, including the earliest industries for agribusiness, astronomical instruments, ceramics, chemicals, distillation technologies, clocks, glass, mechanical hydropowered and wind powered machinery, matting, mosaics, pulp and paper industry, perfumery, petroleum, pharmaceuticals, rope-making, shipping, shipbuilding, silk, sugar, textiles, weapons, and the mining of minerals such as sulfur, ammonia, lead and iron. The first large factory complexes are built for many of these industries. The Roman domestic water system is perfected, with a widespread network of sewers, public baths, drinking fountains, piped drinking water supplies, and widespread private and public toilet and bathing facilities.
Two types of economic systems take root in the Roman world: politically-driven investment by the government bureaucracy and military, which prompt agricultural development and colonization of under-exploited lands, typically combined with the settlement of veterans and colonists, in state colonies and individual land grants in the provinces, building and extension of the road network, the canal network in Europe and the irrigation system in the Middle East, the establishment of an extensive postal system, and the settlement of veterans in state colonies and individual land grants in the provinces; and market-driven agricultural development, involving the spread of advice, education, and free seeds, and the introduction of high value crops or animals to areas where they were previously unknown, the development of an extensive international trade network, and widespread manufacturing. The first market economy and earliest forms of merchant capitalism take root and a vigorous monetary economy was created on the basis of the expanding levels of circulation of a stable high-value currency (the denarius). Innovative new business techniques and forms of business organisation are introduced by economists, merchants and traders during this time. Such innovations include the earliest trading companies, big businesses, contracts, bills of exchange, long-distance international trade, the first forms of limited partnerships, the issuing of insurance, and the earliest forms of credit, debt, profit, loss, capital, capital accumulation, circulating capital, capital expenditure, revenue, cheques, promissory notes, trusts and charitable trusts, startup companies, savings accounts, transactional accounts, pawning, loaning, exchange rates, bankers, money changers, ledgers, deposits, assignments, the double-entry bookkeeping system, and lawsuits.
Formalized educational institutions for legal practitioners begin to transform into a full-fledged university system as they start to provide formalized education and academic degrees into law, medicine, Romanist theology, liberal arts, and natural philosophy. Their curriculum includes grammar, logic, rhetoric, geometry, arithmetic, astronomy, music, architecture, accounting, Romanist theology, medicine, natural philosophy, and Roman law.
7th Century: The Empire gradually recovers from the effects of the plague. Repeated uprisings and rebellions in Ethiopia and India, unsuccessful attempted expansion into Sina, and the first major civil war in the Roman Empire since the 3rd Century Crisis keep the Roman military fully occupied.
Plagues and civil wars somewhat slow down the pace of technological progress, but the country still sees the development of the cross-staff, mariner's astrolabe, stern-mounted rudder, arch bridge, steel crossbow, oil paint, and several improvements in shipbuilding.
The university system spreads throughout the Empire, spurring a heightened empiric interest into logic, mathematics, natural philosophy, and medicine: notable scientific advances of the 6th and 7th centuries include the development of a decimal place value number system and the zero, systematization of arithmetic and algebra, solution of linear and quadratic equations, and those polynomials of higher degree that could be reduced to quadratics through substitution, first developments in differential calculus, the theory of impetus, the first integrated systematization of optics, the development of chemistry, rediscovery of atomism, advances in trigonometry with the definition of the sine and cosine, secant, cosecant, tangent and cotangent, advances in surgery with the standardization of surgical instruments, the development of a mathematical scale to quantify the strength of drugs, and a system that would allow a doctor to determine in advance the most critical days of a patient's illness, the introduction of systematic experimentation and quantification into the study of physiology, the discovery of the contagious nature of infectious diseases, the introduction of quarantine to limit the spread of contagious diseases, and the introduction of experimental medicine and clinical trials.
Mahayana Buddhism spreads to the Roman Empire.
I was referring to the situation in modern times, after modern education systems and mass media kick in. I fully agree that in pre-industrial times the linguistic situation would be pretty much the way you describe it. However, in modern times I would expect somewhat more linguistic homogeneization to develop. I would expect both Latin and Mandarin to become international lingua franca and being taught as second/third languages in public schools all over the world. It is quite possible that Greek, Sanskit, and Cantonese would reatain enough prestige and diffusion to maintain strong diffusion as supraregional lingua franca and/or literary languages. I am a bit more dubious about the continued strong diffusion of more localized regional languages like Coptic and Aramaic, but it's definitely possible. Same reasoning about the more widespread regional languages in the Sinasphere. I would expect the vast majority of local languages largely to die out both in the Romasphere and Sinasphere.