I'm going to use the example of medieval Spain because that's what I know best at the moment and it's super interesting. A Jew in Spain in the years, say, 1200-1400 (before that Spain was mostly Muslim) who wanted to convert to Christianity would be welcomed with open arms by the Church and the king. There were plenty of them, and they were extremely useful to the Church for multiple reasons.

1. There was a big effort by the Church to encourage the conversion of Jews. There was no Inquisition yet, of course, but it was basically a constant effort. In fact, King James I of Aragon used to visit synagogues in Barcelona and preach Christianity during Jewish prayers. So a Jew who decided to convert would be seen as both a validation of their efforts and a successful result.
2. Jews were VERY useful (whether they'd converted or not, but converts could be very useful for this) in terms of academia and translating books from Hebrew to Latin. This was very important because that is exactly how a lot of great works of Greek philosophy and science made it to Europe- they had been translated from Greek to Arabic by the Muslims hundreds of years earlier, Jews in Muslim lands had translated these books from Arabic to Hebrew, and now these Jews who could speak Hebrew and now Latin as well could translate these books. Again, this is something that was not exclusively about converts (there were plenty of born-and-died Jews who did the same thing) but it couldn't hurt.
3. Super important and famous in the context of conversion- they would become part of the effort to convert Jews. Most famously, the apostate Pablo Christiani was a former Jew whom Nahmanides (Moses ben Nahman), a famous rabbi in Barcelona, was forced to debate. (The debate ended with Nahmanides being kicked out of Spain by James I.) These apostates were often EXTREMELY eager to help their fellow Jews see the light, and since they had both extensive Jewish educations and a very jaundiced view of Judaism, they attempted to bring Jewish texts and show how they actually proved the correctness of Christianity.
4. Also super important (though more in France than in Spain, IIRC)- they were very helpful in "rooting out heresy." Essentially, they would go through all of their old Jewish texts and try to find "heresies" which criticized Christianity. They would present these to the king, and the king would use this as grounds to burn Jewish books (something which famously happened in Paris with the Talmud in 1242) or even expel the Jews (which happened in France in 1306, among other times).

What did the Jews think about all this? They were not happy, understandably. In theory they would ostracize the apostates to the extent that was possible- it wasn't always possible because the newly Christian apostates were often considered of a higher status than them and they didn't always have the right to shun them fully legally.

I'm not going to get into 1391 and on (basically the beginning of the Spanish Inquisition era, though 90 years before the Inquisition was actually established) because the dynamic was QUITE different. If you're interested, though, I'd be glad to go through that.

EDITED: grammatical ambiguity, wrong year, wrote Judaism instead of Christianity at one point

Also, I will mention- #2 above is probably more of an issue a bit later later than it is in 1200-1400. It becomes really relevant in the Renaissance, and is especially relevant in Italy. But it's still a relevant point in general.