Once inside London travel zones, basically Center of london and surrounding boroughs, then kids are free. So best option is to buy the weekend travel pass for the london zones. It wont cover you from the airports, at least not Gatwick, Stanstead or Luton. I think heathrow might be covered. But just pay seperately for the return to/from airport and then get your weekend travel pass for jumping on/off tubes, trains, buses in and around london.
Their public transport has gone crazy expensive, but this is the best option for you.
I was their a couple months ago, flew to gatwick, and this is was the best option i found.