Have Christianity fragment in Northern Europe, preserving some regional languages due to being associated with certain sects (which no authority is able to eliminate). Perhaps the Noric language could survive this way, along with some East Germanic languages. IMO some assimilation of West Germanic speakers is essential for East Germanic languages to have a reasonable footprint.
The Middle East fragments into many languages. A unique Coptic language in Sinai, many Aramaic languages in the Middle East, a few divergent Greek dialects along the coast (and the unique Crimean Greek language).
More people migrating into the Tibetan/Sichuan/Yunnan Plateau. This hugely diverse region of the world could be a refuge for linguistic groups. The example of the Bamar, who originated in this language yet nowadays are dominate over much of Myanmar, is telling. Imagine if instead of the Bamar, it was a group of Tocharians instead? Although this would be very hard given a 1 AD limit, it's perhaps possible to get some Indo-Europeans in those mountains, even if they aren't recreating the success of the Pagan Empire. But if they weren't as successful, perhaps Burma would have Tocharian (not called that of course), Bamar, and Pyu peoples in addition to the Mon people of the South.
No Islam is key, but also the weakness of Christianity. Different Christian sects could preserve the Elamite language in Persia, at least some of the Caucasian Iberian languages, and some Anatolian languages (like Isaurian) in addition to likewise preserving the regional diversity of Indo-European and Caucausian languages alike. If the Tocharians don't migrate into the mountains of Central Asia, then Tocharian-speaking Christians could be a thing.
Malta and surrounding islands could easily be Punic speaking to this day, assuming different migrations in Late Antiquity. Although OP didn't specify North Africa, Malta could also speak an African Romance language akin to that of Carthage (perhaps speaking both a Punic language and an African Romance language!). No Islam means Hispania will be very diverse in terms of Romance languages, along with Sicily and the Balearic Islands, and places like Andalusia, Sicily, and the Baleares will speak Romance languages close to what's spoken in North Africa (and Sardinia). But indigenous Iberian languages could remain given fragmented Christianity, since it might let the Cantabrians and maybe a few other Celtic groups remain in Iberia.
Steppe migrations cause permanent establishment of new languages. Cuman survives in Cumania in Hungary but especially in Transylvania. Tatar languages are established as minorities throughout Europe.
Scotland could be linguistically divided between Picts, Cumbrians, Scottish Gaels, Norse-Gaels, pre-Indo-Europeans (on St Kilda?), and Scots speakers. The Faroes and/or Iceland perhaps could retain a population of Old English speakers relocated by the Norse, in addition to hypothetical settlement by Sami thralls (why not move some Sami there to tax them?) or even the Inuit/Dorset/similar culture.
Overall, linguistic diversity in a region as vast as Eurasia with as much recorded history is a huge question.
central asia was always very diverse - I'm wondering about the more heterogeneous parts of the world (at least in the modern era) - china and northern europe, mainly
Finno-Ugric languages (mostly Samic languages and the ancestors of Baltic Finnish languages) had a wider footprint in 1 AD than in later centuries--maybe some patches of Samic speaking peoples in Southern Sweden or the mountainous parts of those areas?
A POD at Jesus's birth is a lot of time, but I can't imagine the Finns doing better than their southern neighbours (Balts and especially East Germanics and Slavs). The best a Finnish tribal group could do (call them the "Tavastians" or "Savonians" or something obviously Finnish) is give their name to a region populated by speakers of some other language, aside from a scenario involving the mountains of Europe where perhaps these Finns who gave their name to the region also survive in language, which would be obviously Uralic like Finnish. In that case, they aren't much different than a hypothetical survival of the Noric people or other continental Celts. If the Hungarians could do something similar, then I don't see why an early enough POD couldn't allow the Baltic Finns to do so (although for maximum Baltic Finn wank, a Finnish North America is a possibility as late as the Kalmar Union).
Would they sound like northern Kyushu dialect?
Definitely not the Kyushu dialect of the Meiji era and beyond.