Will help you out with one tip regarding transportation. For transportation for the adults, just use a contactless credit card or your phone (Apple or Google Wallet) to pay for the Tube/trains/buses. The 8 year old (and any kids 10 and under) can ride pretty much everything for free- Tube, buses, Elizabeth Line, Overground.
For the kids 11 and over, buy an Oyster card at a ticket machine at Heathrow for 5 pounds for each kid. Then go to an agent and have the agent put the "Young Visitor Discount" on each Oyster card, and that will make all fares 50% off. Then put say 20 pounds on each card and you are probably all set for the entire trip.
https://www.londontoolkit.com/briefing/london_transport_child_fares.htm
Once you get to your very final destination in London (sounds like maybe St Pancras for the train to Amsterdam), after you exit the Tube for the last time, go to a fare machine and you can get a cash refund for any fares remaining on the Oyster cards. Unfortunately you can't get the 5 pound original price back. You can then cash in those pounds for euros at a bad exchange rate at St Pancras!