You also have to keep in mind that, at the time, Europe was largely considered "barbaric" by the Romans (and Greeks before them) and most major trade routes went east-west; Christianity spread along the lines of least resistance in these cases, reaching India and China. India's proportionally small Christian population does pre-date Christianity in Europe, as does Ethiopia's embrace of the same religion. Essentially, these regions had closer ties to trade networks which linked them to The Levant (where Christianity started), but Europe didn't have such links.

Christianity's expansion into Europe, by comparison, was relatively slow - almost stately and hit-and-miss - with Christian European kings like Charlemagne still warring with Pagans like the Saxons in the 8th and early 9th century. Also, the Norse were largely pagan well into the 10th century.

Part of your disbelief, I think, comes from a skewed telling of Christianity and the importance of Europe in the pre-modern period. Non-Mediterranean Europe was relatively poor, relatively unenlightened, fragmented, and dangerous during the late classical and early medieval period while those cultures further east were pretty stable, comparatively. During the later part of the modern era (say the 18th and 19th century) when Europe was becoming an obviously dominant region, able to extend its power and influence to all points of the globe, the natural assumption was that since Europe was dominant and sending missionaries to India, China, Africa, and so on, then Europe must be the cradle of true Christianity (and to a larger extent, the cradle of 'civilization'). It is a belief that was accepted in the same way that Social Darwinism was or that colonialism was; pseudo intellectuals accepted it as a fact that was both natural and indisputable (despite the fact that it was completely wrong and ignores evidence to the contrary).

It is very hard for Westerners to alter these prevalent, colonialist interpretations of the world without serious soul searching and critical thinking; I'm glad you came here to ask this question instead of ELI5 or AskReddit and I hope my answer helps clarify a bit.

You have to realize that Christianity didn't start in Europe, it started in the Middle East, namely the Levant.

Antioch and Alexandria where early centers of Christianity before Western Europe ever was, and various early missionaries spread it as far as trade routes would carry them. The trade routes reaching from the Roman Levant to India where very highly developed, with the Parthains being the only major power in between Rome and India.

Christianity was very widespread throughout the East, and given that Israel would have been closer to Persia than Rome, its not hard to imagine that Christianity would have spread there. Indeed, most of the missionaries to China and India (both had ancient Christian populations, neither of which had European roots) originated from the Church of the East that existed in Persia under the Sassanids.