Maybe an example of leaving it to FT.

There's no problem. Aircraft are full of aunts, uncles, grannies taking young relatives with them. In many parts of the world, grandparents are the default carers while the parents work, and that's well understood by airlines. Airlines are also well used to people cancelling flights for one person out of a party. On domestic they don't even check for ID if you are going hand luggage only and have online checked in.

Out of LHR your children should have passports with them if at all possible, but not for any of the reasons mentioned in this thread. The photo scanner is notoriously unreliable for children's faces so a passport will minimise trouble at the gate if the double red light comes on. Not an issue in the other direction.

Internationally it makes sense if the children have a letter from the principal carer authorising travel with someone else, particularly if the children are female, with telephone numbers on it. But if the travel is with someone sharing a family name it would normally not be an issue. But occasionally UK Border Force check up on it, which is one reason why children can't use the e-passport gates.