This is an interesting question; I figure I'll share my opinions first from my own perspective, and then I'll take some stabs at guessing how the wider Church would respond.
I'm not necessarily opposed to the idea of Jesus's excursion to India, but I find it extremely unlikely. I'm not quite sure what evidence you'd have to give me in order to believe these claims, but it would have to be quite substantial. I've studied numerous differing accounts of the "Historical Jesus" and am quite aware of how limited our knowledge of "who Jesus really was" is, even according to the well-established stories of the Bible.
In case you're unfamiliar with the "Historical Jesus", it's essentially a movement to identify how much of the New Testament accounts of Jesus are historically accurate, and how much have been generated many years afterwards with specific purposes in mind. In a way you could view it as an attempt to separate the Jesus of history from the Christ of the Bible. Scholars from all over the spectrum of Christianity (from Liberal to Conservative) participate in this movement, and use the whole gamut of scholarly criticism from analyzing the structure of sentences in the Bible to running soil samples. The more liberal scholars will downplay or outright eliminate the Christ of the Bible, while the more conservative scholars will find ways to marry the two, or at least reaffirm that the Christ of the Bible is still essential to our modern faith, but I think it's safe to say that all agree that the two are not identical. Note that we're talking
scholars here, not laypeople or theologians.
One of my favourite books on this topic is
The Historical Jesus by John Dominic Crossan, in which he collects every event or saying involving Jesus from the four canonical gospels as well as the Gospel of Thomas and the hypothetical Q document (it's accepted by over 99% of biblical scholars that Matthew and Luke both share as a source Mark and some long-destroyed "collection of sayings of Jesus" document which they dub "Q"). Anyway, Crossan collects these verses, groups them according to the number of independent documents they occur in (ie. if the essence of a verse can be found in Mark and Thomas, both of which were written completely independent of each other, then it's counted twice), and finally sorts them according to the number of occurrences. Verses with a higher cardinality are considered more likely to be historically accurate, and verses with a cardinality of 1 are essentially discarded as being later additions. Crossan then analyzes the result and draws inferences using cultural and historical anthropology to connect the dots about who Jesus likely was.
Anyway, the reason I bring all this up is because your question falls squarely in the Historical Jesus umbrella. And the reason I've described things in such depth is to demonstrate how controversial the simple question "Who was Jesus?" really is, and the extremely scientific and systematic way the question is approached. There is very little about Jesus that is fully established and accepted among the best and brightest scholars who have dedicated their lifetimes to the research. The discovery of some Jewish woman's body and a genetically similar crucified man, even if they were established beyond reasonable doubt, would still be very thin evidence of
actually being Mary and Jesus. As Reverend Richard said above, crucifixion was a very common practice at that time.
The idea of Jesus living in India seems to me to be mostly sensationalism, and is found here primarily on the same television stations that "prove" aliens built the pyramids and sunk Atlantis. (I know because my dad unfortunately frequents those stations.) As far as I know, scholarly consensus on the idea is almost unanimously against it.
But say, hypothetically, that it could be proven, and that scholars were suddenly all, "Hooray we all believe now that Jesus lived in India now!!!!!" Then I would totally jump on that bandwagon.
Moving beyond myself to the wider Church, you'll encounter all kinds of opposition. The more liberal denominations are strongly influenced by Historical Jesus scholarship, so if you convince the scholars you're likely to convince them too. Theologically, the resurrection of Jesus would need to be reinterpreted, and would probably end up more metaphorical (which is how I currently interpret it anyway).
For those of a more conservative bent, theology usually takes precedence over scholarship, and it would take centuries for the idea to be fully assimilated. Many for whom Jesus' resurrection is ultimately important and who struggle with metaphorical reinterpretation would lose their faith. Biblical literalists would continue to deny the evidence with claims like "God must have planted the evidence to challenge our faith," which is how they currently explain dinosaur bones.
...And of course war would break out and Planet Earth would be destroyed in a nuclear holocaust which would reverberate to the edge of our solar system and God would look down angrily and shake his fist saying, "I hope you all burn in hell, you heretics!"
