A the link shows, the simplest way to travel around London is for the adults to use contactless debit cards. (You 'tap in' and 'tap out' at Underground stations. You just 'tap in' on buses). There's a daily cap on what you pay with a contactless debit card, which is exactly the same amount as the daily cap on Oyster cards. (The actual amount at which the cap cuts in depends upon the times of day during which you've travelled and which zones you've travelled through).
At Underground stations the group should use the wide barriers, to one side of the main ones, so that the children can pass through without any form of ticket. (Those barriers are always staffed, so it's very easy to do).