Can I ask how are you checking the seats and that the flight is nearly full?

Are you using BA's "manage my booking" to see which seats are available to purchase? If so, be cautious about drawing conclusions from that because it won't necessarily show you every empty seat. As I note CatchThePigeon has just said, it's very possible that the system has already allocated some seats for you.

You can use expertflyer.com to see a truer picture of which seats are occupied - I think there is a free tier which allows seat queries or if you are happy to post the flight specifics, someone here can certainly look it up and post the seat map to see if it corresponds to what BA are showing you.

Nonetheless, even if the flight is actually full, and most seats are actually reserved (which is certainly possible), BA can and will sort it out to meet their guarantee, even if they need to move people around on the day. Just for info, here is their policy which confirms that your children will be seated adjacent to at least one adult
https://www.britishairways.com/en-gb...travel/seating

Hopefully once the free seating opens up you'll be able to get something suitable, but do rest easy that BA will ensure that you are not completely separated.