Just got home after a week in London where I entertained the LO while DH worked. Here are my tips:
Try and take the Uber Boat / Thames Clipper for a great combination of public transport and sightseeing. You don’t get any commentary, but it’s a lot cheaper than doing a tour. It’s cheaper if you use contactless than buying a ticket - think it’s £7.30 per adult in the central zone which runs from Tower Bridge to Chelsea or something similar.
Do note that the boat is not included in the TFL daily travel cap. I think the cap is set at £7.40 per day, so that’s the maximum spend using contactless no matter how many trips you do. It generally works out much cheaper than a travelcard, but check the tfl site to double check. Once I realized, I would jump on a bus even for a just few stops to save the endless trudging.
+1 for the Science Museum. The really interactive stuff is in the basement, with a fun exhibit of home items with lots of buttons and lights to push. There’s also The Garden area for 3 - 6yo with water play, fun mirrors and lots of things to push and explore.
The Natural History Museum is right next door to the Science Museum, and it’s currently holding a HP Fantastic Beasts exhibit. It’s ticketed and not cheap, but may be worth checking out if you have HP-mad kids.
The Harry Potter Platform 9 3/4 is a good photo op if you’re in Kings Cross, but it’s not really worth trekking across town for IMO. Also beware the enormous HP gift shop next door if your kids are souvenir hunters!
The Thames Path is now really good, with lots of waterfront cafes and green areas. We walked along from Waterloo to Tower Bridge (about 40 mins), passing The Globe theatre, The Golden Hinde (maybe good if your kids are into pirates and adventures), The Clink jail, and some lovely open green space for running about.
Finally, the new Walkie Talkie tower near Monument has a free to visit Sky Garden on the 30-somethingth floor with amazing views. Book a few days in advance and hope for clear skies.