There is no option to stay 'airside" in either direction. As long as as you are in the queue for security to get back airside within 30 mins of ETA you will be fine. You could cut in finer, but not with children.

With only 2 hours o/b you will have to take your pick of the concessions. The most likely is the airside Italian theme place. Not too expensive. Quick. Good for what it is. Better than the sandwich places (there is no Pret i don't think)

On the return thats way too much time to stay entertained at LCY. I'd go up to Stratford. Loads of food options in the Westfield centre, and you can look around the Olympic park. Canary Wharf area is also an option - there is a Zizzi pasta / pizza place overlooking the Thames at Westferry Circus (ask for the Four Seasons hotel) with loads of outside seating. Either way - with a family - a Taxi wont be too different in cost to Public transport and more more convenient.