I mean, the Italian peninsula and Greece are in Europe, and there are New Testament epistles addressed to communities there which are thought to have been composed by the second half of the first century. So no, Christianity didn't get to India before Europe. That said - and here I'm leaning on J. Philip Jenkins' Lost History of Christianity, particularly around pp. 64-66 - it certainly seems fair to say that it spread eastward at least as rapidly and early as it did to the west. The local tradition of an unbroken lineage to Thomas is probably beyond the scope of what historians can verify, but Jenkins puts the appearance of Christianity in southern India "no later than the second century", a couple hundred years before its legalization in the Roman Empire. Meanwhile, in China, the "earliest formal mission can be dated to 635", which would be roughly contemporaneous with the Gregorian mission to the Anglo-Saxon kingdoms. So even if they aren't attested as early as Christian communities in the European Mediterranean world, those in Kerala are at most not far behind, and there was missionary activity even farther east that predated the conversion of decent chunks of Northern Europe.