I lived in London for six years and worked transatlantic flights.
For such an early arrival, you might to chance a nap. I would make it no more than 2 hours and set your alarms! Do not sleep past noon and be sure to get your heads on the pillows as soon as possible. Toss out this idea if there is a delay and you arrive after noon.
Please do NOT do any kind of tour of anything you can't "get out" of in a hurry. Try to see outside sights. Walk around Leicester Square or Covent Garden. Watch street performers and be around awake people. They wont appreciate any museum or tour. They'll just zombie-out or feel ill. Go see some of the amazing parks... Drink, walk, get out in the fresh air (if it's not raining!)
By evening, you will be feeling better but don't be tempted to stay out late. Plan to be in the hotel or whatever by 10pm. Indulge after the first night.
I did find Melatonin helpful and your children might be old enough to try it too. Do some homework and read up on this. I have many, many drug allergies and since this already exists in our brains, I had no problems taking it.
Don't fall asleep on any public transport! Pickpockets abound.
This pattern of TATL flights landing at 7am is starting to get irritating. I'd like to see them arriving at 12pm (local), to help with the time change (if the gateway city is your final destination). A 10pm departure should be able to do a 12pm arrival, which works better to fit better into the hotel checkin cycles
Interesting suggestion but actually, London is a big connection point so the East Coasts flights come in early to get those continuing on another flight off smoothly. There's kind of a flow to international travel, which includes the fact that N.America-Europe flights arrive in the morning...