The choice comes down to whether there's one or more of the object in question. The children aren't all following one nose, right? Each child has a nose to follow. So, the children follow their noses.

The other sentence, however, could go either way. If the children share a house and they're all going back to that same house, it should be singular:

The children went home to their house.

If they are going to different houses, it should be plural:

The children went home to their houses.

If the type of dwelling isn't significant, you could avoid this conundrum all together with a more direct sentence:

The children went home.