That's a much more complicated question. To give the shortest possible answer, the new religion usually incorporates aspects of the old one, sometimes on purpose, like Christmas (pagan festival of lights) to ease the transition of conversion. The threat of death can also be very compelling. Each example in history was a bit different, so its difficult to provide a good answer.

If you are very religious you might view your defeat as a sign that your god has abandoned you or that their god is more powerful. Good reasons to warship someone else. If you are not, you might just not care that much either way.

There is also an element of Stockholm syndrome as well. Not forgetting another religeon sweeps in often by force, clearly they are more favoured by the gods and your god wasn't real or didn't care.

Also keep in mind that ancient religions don't always match cleanly with what we expect out of modern ones.

Especially the idea of deific exclusivity. If you have always believed that your gods and your neighbors gods are different and war with one another just like you and your neighbors do, then maybe they lost along with you.

Plenty of stories about gods dying, probably feels better that your gods died trying to save you than ran off and abandoned you.

Just one speculative example, but no one today would assume that a personal problem is because their religion got annihilated, maybe not so much with all the numerous past belief systems.

They don’t have to. They just need to act like they do or get killed. And teaching your kids the old religion would see them get executed so you don’t teach it to them. Your kids only get instructed with the new religion so within a generation or two, the old religion dies out.