Well here is Britannica-1, and oddly the only 21st century Britannica as well. From Steve Jackson's description, the Colonies fractured after independence which allowed the British Empire to simply dominate in North America, and effectively everywhere else. They defeated Napoleon, won in Crimea, Fought Germany twice, and have been the world's superpower since the 1850s as far as I can tell. I was interested in Britannica-1 because it is the only post-colonial Britannica in the lot. Unlike my previous maps I am just going to write a paragraph about each region.
In North America, the British reign supreme. The Dominion of North America stretches from the sleepy fishing town of La Paz all the way to the Newfoundland coast. Following the "Claims War of 1785" the UK took advantage of the former colonies infighting and simply annexed the region that was the Northwest Expanse. From the 1790s to 1810s there were many attempts to grab more of North America but it was only after Napoleon's defeat that the Louisiana Cession saw the territory added to the empire. Texas is theoretically an independent nation as a protectorate but a member of the Commonwealth of Nations. California was sold to the UK to pay off a series of debts accrued by Mexico. The "Seaboard Colonies" are a series of nations well known as a collection of tax havens, lax regulation or radicalism. The Commonwealth of Massachusetts is actually a member of the Commonwealth of Nations, and regarded as the most normal of the former colonies. New York has emerged as a financial center and tax haven, it's a question how much wealth bound for Inland Revenue is in some house in the Hamptons. The United Federation of North America encompasses New Jersey, Pennsylvania, Delaware and Maryland, and their 1784 Constitution holds the record for the longest in force in the world. The Virginian Commonwealth has in recent years (since the collapse of the agriculture market in the 1920s) as a tourist stop, the derbies in the Western section of the state attract even royal audiences. The old "Antebellum" charm of Arlington captures worldwide audiences following a writers series of alternate history novels about "The Federal Republic of Columbia". The Kingdom of Carolina is an anomaly as it's reverted to a monarchy following a declaration of the Governor in the 1860s. They hold the ignoble distinction of holding onto slavery until the 1910s. The Colony of New Lindisfarne, established in 1850 features the autonomous zone of the Seminoles, but a tropical destination for many in the Bahamas. Mexico has had a very troubled history, as it was a French puppet for 40 years in the 19th century.
South America followed much of OTL's history except that Bolivar was able to hold together the Latin American countries, at least some of them.... Well only Colombia and Venezuela. British interests were only attracted to the continent when in 1964 Gran Colombia fell to Bolshevist guerrillas, backed by Colombian farmers. Besides a minor war with Argentina in 1979, the UK has had little interest in South America.
Africa, was sliced apart by both France and the UK, and is still home to several European Colonies. However where the French have had nothing but disaster, the British have had success. The East African Federation, is to varying degrees a neo-colonial state, as British commercial interests dominate the region, but large scale infrastructure, education and investment in local business firms has made the region developed, and an important member of the Commonwealth. However one could say the British were lucky as Congo Bolshevist guerrillas attempted to disrupt the Federation but only really galvanized the Federation's population against communism. The Dominion of South Africa, never fell under the sway of the National Party (outside of a 3 year stint in the 1910s). There are still racial problems but following Acts of Parliament in 1897 the "Diamond Act" (also known as Rhodes pissed off the Colonial Office Act) which saw the creation of the modern Dominon of South Africa and decades of careful management, tensions are not as high. The Congolese Bolshevist Union is a remnant of the old Bolshevist War, but is emblematic of the problems of Bolshevism: corruption, mismanagement, believes its own propaganda and dictatorial.
The Middle East is home to several Muslim kingdoms, and backed by British Arms and the occasional military base. There is a reason the British Admiralty has a carrier group in the Arabian Gulf, and a mistrust of BP (responsible for the Mossadegh debacle of 1959, and the criminal mismanagement of the Colony of Lagos & Ibadan). India was at one point a Dominion but during the war with the Soviet Union (1958-1963), the subcontinent fell out of British hands. A special conference in 1961 quickly set the borders that currently exist, and it's a miracle war hasn't broken out yet.
East Asia is one area where outside of the Hong Kong, the Foreign Office has little to no influence. The Republic of China, still despises the terms that Eden forced them to sign in 1948 for arms (which gave a 50 year extension on Hong Kong), and the United Kingdom's lack of any meaningful support against the Bolshevists on the Asian front (1958-1963). The Empire of Japan is even less receptive to the British, as a minor war in the 1940s led to a national disgrace for Japan, and continued British domination of Oil prices in the world. Oddly Japan at the Hiroshima Nuclear Institute is developing new types of nuclear power generation and Energy Storage. However Japan's Empire is still very much trapped in the 1940s, and no nation is willing to challenge their current territories, save China. Korea was the site of a horribly long proxy war between Japan and China in the 1960s, which resulted in Japan losing it's colony and only ingress to the Asian mainland.
Europe, is still the home of the Colonial Powers. France has been a somewhat ally of Britain, and supported Greek Independence when Constantinople fell. Germany retains Danzig and Silesia, although they lost Konigsberg to the Soviets in the Post War conference of 1941. The most recent news in Europe is the collapse of the Soviet Union in 1993, and the establishment of the capitalist Russian Republic. A new age of British Dominance is upon the world. With an absolutely insane level of stable colonies, a navy that keeps the seas safe (save the North Pacific), and above and beyond the largest economy on the planet, Pax Brittanica is upon this world, soon a British Cosmonaut will land upon Mars.
However oddly Infinity's probes have not found any Centrum operatives, and it took Infinity to figure out why. Buried in an academic database at Cambridge is a scientific journal article, that is the theoretical basis for parachronics. Infinity and Centrum both have no idea if Britannica-1 has or has not developed parachronics at all.
Yeah, although Britannica-1 is a Brit-wank, it does have its dark spots. Imperial Japan still exists, and racial struggles are still very much present in Africa and India. Europe's military Alliance is failing, because many European nations see the threat is the UK more than any other. The Commonwealth of Nations, and the Imperial Parliament is dysfunctional at the best of times. It may be Pax Britannica... but if OTL is anything to go by the 2000s are not going to be kind to Britain. In the end, it's a post-colonial Brit-wank, where you can find the romantic angles if you look, but there is a bit of darkness that would the extrapolation of some British Colonial policy even if it did function perfectly.