I got quite carried away researching this one!
The short answer: get a Family and Friends Railcard and buy Anytime Day Singles Thameslink only from Gatwick to London Terminals. Travel on the xx:06 or xx:36 Thameslinks from Gatwick to St Pancras, and change to the Victoria Line for Tottenham Hale (using Contactless/Oyster from St Pancras). When returning, you can travel direct from Tottenham Hale and buy, with you Railcard, Anytime Singles to Stansted Airport.
The longer answer:
From Gatwick to London your options are Thameslink, Southern, or Gatwick Express. There's no particularly good reason to use Gatwick Express - it's more expensive and only about 5 minutes quicker than Southern's services to Victoria. Southern will take you to Victoria in about 25-30 minutes whereas Thameslink is a bit more varied - between 30-45 minutes to the first central London stop, depending on the stopping pattern through south London. If you're going straight to Tottenham Hale from Gatwick, then the absolute fastest option would involve taking the faster Thameslink services to St Pancras, where you can pick up the Victoria Line tube to Tottenham Hale. As the name suggests you can also get this tube from Victoria station having taken the Southern service, but the time it takes to go through central London on the tube more or less wipes out the slightly faster journey into London from Gatwick. The exact fastest at any given moment however depends on when precisely you arrive at the station at Gatwick as services are not equally spread through the hour: sometimes its better to wait 5 minutes for a quicker train, at other times you're better off just getting the slow train as the wait at Gatwick cancels out any time savings. The obvious ones to avoid are the xx:03 and xx:33 past Thameslinks as there's a quicker Thameslink leaving at xx:06 and xx:36 past the hour.
The adult off-peak cap for contactless journeys including Gatwick is £19.90 (note that this does not include Gatwick Express). The off-peak cap for a day's travel in London zones 1-3 (Tottenham Hale is in zone 3) is £8, so your journey from Gatwick is effectively costing you £11.90. This is marginally more expensive than the cheapest available ticket for Gatwick-London, which is £10.70 for a Thameslink only Anytime Day Single. For children, the 11-15 Oyster brings much lower caps around London, just £1.50 for the day off-peak in most zones. However for journeys involving Gatwick the children's cap is £9.90 for the day. A Thameslink only Anytime Day Single from Gatwick to London Terminals is £5.35 for children so there's more of a saving in getting that and then using Oyster for journeys once in London. If you get yourself a Family and Friends Railcard, then the combined cost from Gatwick to London Terminals Anytime Day Single Thameslink only is £18.90 so that's quite a saving overall - you could consider this equivalent of bringing the adult fares down to £4.10 per person. The relative saving of the Thameslink ticket with Railcard obviously reduces if you're travelling outwards to zones 4-9 after going to Tottenham Hale - the further out you travel, the higher your cap and so the cheaper the contactless option from Gatwick becomes, but I think the paper ticket with Railcard is still going to be the cheapest. It does reduce the number of services you can use from Gatwick of course but there are still 6 Thameslink trains an hour.
Travelling to Stansted is simpler. Conveniently, the vast majority of trains from London to Stansted call at Tottenham Hale, from where it's £16.20 for an Anytime Single per adult and children cost half of that. That's a combined cost of £48.60 or, with the Railcard, £27.60. If you're in central London for the day, then trains start at Liverpool Street and the overall cost is only a couple of pounds more than travelling from Tottenham Hale.
A Railcard is £30 but look around for online discount codes - you can usually find £5 or £10 off. If you have Clubcard vouchers spare, you can use those to buy a Railcard for £15 worth of vouchers.
