How can I automatically sort adult villagers from babies. once they've grown?

If you make a water elevator with signs, baby villagers can't climb the elevator, so they are forced to grow up into adults before they can interact with the elevator.

So if you make an elevator inside the hall, it should block the babies from escaping.

I can't guarantee it will work on Bedrock - best test it in creative first - but Gnembon has developed an ingeniously simple and easy to build device that holds baby villagers until they grow up.

This is the device as implemented as a part of his villager breeder:

and this is the same, stand-alone and with side walls glass replaced with barrier blocks. Buttons were placed behind water source blocks - all remaining water is flowing.

The idea is ingenuious in its simplicity. The short bubblevator (with the standard push-pull features for pulling mobs into the water column - a wall block, and a gap for water to flow in) ends with a glass block 'cap', so any babies will get trapped under the glass block. (don't worry, they won't drown - bubble column provides air.) The moment they grow, their feet remain on the same level, but their heads glitch into the glass block above - it stops blocking them, so the bubblevator is able to push them even higher. Then they are caught by the water stream above and pushed through the open fence gate into the collection area.

Make a structure like this: Underside:

The benefit of this is that if for some reason an adult comes through, it will trigger immediately.