I cannot doubt for a second that BA would love this family. 5 passengers in F (paid or award - who cares) is a big deal compared to a few complaining passengers. In a few years, the kids will calm down and settle down and they'll be great life time customers.

Sure, the parents could have been a little more considered and seated themselves closer to the kids and helped calm the kids down but sometimes people just don't think much about it.

There is no way BA is going to call them and "warn" about anything.