Bugger - just wrote loads and then accidentally shut the window!
Interrail ticket - 2 adults 2 under 12s - between 970 and 1200 euros to cover 2 weeks.
Eurostar seat reservation ON TOP - 32 euros EACH - robbery - and that's each way. So 248 euros
So:
London - Paris - 200 euros - but you might get a better bargain.
Paris - Marseille - Ouigo.com - currently as little as 22 euros each less for the kids. Can't look at next summer - but say 100 euros.
Marseille - Lyon - as little as 26 euros and 14 for a child - 100 euros
Skip this bit:
Lyon - Geneva - 29 euros for an adult 14 for a child - 100 euros
Or you could skip Lyon and do it in one go for slightly less. I'm off to Lyon next month though and it looks nice :)
Geneva - Amsterdam - bit of a bugger. Geneva is a bit of an outlier on the train front. Easy from Lyon but not anywhere else.
Lyon - Amsterdam - 100 euros adult 56 kids. So 300 euros - sure could do better than this - change in Brussels.
Amsterdam - Brussels - For everyone 112 euros 2 adults 2 kids (kids under 12 free on Belgian trains)
Brussels - London - about 200 euros.
Eurostar is merging with Thalys in France this month so there might be interesting time table changes and different direct routes!
So with careful booking London - Paris - Marseille - Lyon - Amsterdam - Brussels - London I just found for about 800 euros. Seat reservations would be on top, but you'd also need to pay those for interrail bookings if you wanted to guarantee seats together. So say 20 - euros at least per train (not the Eurostar where they try to rob you) Not really needed for regional train between Brussels and Amsterdam, only InterCitys in France.
For Lyon, you could swap Avignon, or Aix en Provence without making too much of a difference I think. Better for swimming pool options. The trains pass those places. Geneva is a bugger though and will cost you loads more. In 2 weeks I think you'd have plenty. For accommodation I normally use Booking.com - plenty of choice of hotels and apartments.
These are just random dates so only representative - best deals when tickets first go on sale. In France it's 3 months ahead.
Phew. Hope it's useful.