If BA wants to charge for seat selection, it really should have implemented systems to avoid this kind of thing happening. It is blindingly obvious to the universe that people pay for seats to be sat together and/or to obtain a preferred seat. Keeping the money and separating pax or moving an emergency exit seat to a random row is NOT what people were choosing to pay for / using their status for.

First of all, call and insist upon a refund for the cost of seat selection. You paid to avoid this aggravation and you are getting it anyhow...

Secondly, depending on the age of your kids I would remind the agent that minors need to be sat with their parents, not split across the cabin randomly. If the kids are young enough, then it's a legal requirement for each one to be sat next to a parent/accompanying adult. So the plane won't fly as seats are currently allocated anyhow. Therefore two window + aisle combinations will need to be found. If the kids are old enough, then I might relax, insist on the seat refund, and then wait and see whether the trip actually happens.