I think that a "conservative Beatles" might work better if they were sort of non-political and their conservatism was mostly social. Not exactly "kill the commies" in their songs, but they stick to their early style more, and their subject matter is less subversive.
Americans would see them as charmingly old-fashioned and English, while English people would probably not mind terribly if they sort of stayed out of politics and were seen as playing decent songs and being a wholesome example for teens who liked their music.
I would imagine they would be less popular than they were in OTL, but could still be reasonably well-liked. Their lack of status as counter-culture icons would probably protect John Lennon from his murderer's obsession. Thus, I imagine the Beatles still break up eventually, but re-form from time to time with relatively little fanfare.