How long does it take for women to give birth? I have 2 women pregnant right now. Can I send them off to work in other rooms or do they have to stay in the living quarters until they deliver?

3 hours to drop a baby and another 3 hours for the kid to grow up, don't bother giving pregant women weapons as they just run away when bad things happen.

Can I send them to work in other areas of the vault or do they have to stay in the living quarters until the kids are born?

They can work, but IIRC they take a bit of a happiness hit. They will not respond to incidents and will in fact flee to another room when it happens, even if they're one of your top dwellers and are a main forager with a top-shelf gun.