Hello I hope you have a fab trip to London.
I’d forget about London Dungeons unless your children love life-sized plastic models of people torturing each other (think moving models ripping out intestines and roasting them in front of the still-living victim) while a sadistic actor tries to upset the kids and keeps saying “this really happened!” I found the place absolutely disgusting as a teen.
London Eye is fabulous, the museums (particularly Natural History Museum, and Science Museum) are wonderful. The Shard and St Pauls Cathedral also great as is Harry Potter. The best way to get around will be tube. I strongly recommend you don’t drive at all in London except to Harry Potter Studio and allow a lot of time as people drive very aggressively in London and navigating London while watching the drivers is exhausting. London is huge: there are more people in London than in the whole of Scotland. Use google maps directions function to measure walking distances.
If I was staying in Islington and taking a friend to London for the first time then I would…
DAY 1 get the tube to London Bridge, have coffee in the Shard cafe (amazing views), walk to London Eye and do that, then get a speedboat or Thames Clipper to Embankment, grab lunch there (eg pizza express near Trafalgar Square), walk up to Buckingham Palace via Trafalgar Sq and St James Park (pop into National gallery at Trafalgar to see a Van Gogh), get tube from Victoria to Oxford Circus, do some shops including Hamleys, then tube back home.
DAY 2 drive to Harry Potter Studios, spend morning there, drive to Legoland afternoon there if have energy, drive home.
DAY 3: tube to National History Museum, walk to Science Museum next door, soend morning there and perhaps walk up to Hyde Park. Then get tube to Covent garden, have lunch and do shops there (watching out for the pickpockets!) then walk to Back to the Future musical which is epic and perfect for children that age. Then is only a few minutes walk to Charing Cross station from where tube home.
Obviously adjust depending what cjildren are interested in.