Is riding the subway (tube?) easy to navigate?
How about Scotland and Ireland? Will have about 10 days.
Appreciate all the advice for sure...
In London, make sure to stay at a chain hotel, don't trust bookings.com reviews and photos. There's a metro line just below Heatrhow, once in Euston Station you can buy a day rider bus ticket that covers the whole city for around £6 per person.
Other places to visit:
Chester, a pleasant city with Tudor architecture near Liverpool where you can walk on top of a medieval wall that surrounds the center, there's a fast direct train London-Chester that gets you there in 2 hours.
Liverpool, maybe you like the Beatles... Chester and Liverpool are on the same rail network, for about £6 you can ride all the bus/train/metro/ferry.
Dublin, maybe you like Guinness? A Ryanair round trip ticket from Liverpool is about £20. Beware that Dublin is even more expensive than London.
Bowness-on-Windermere, city in the Lake District, preserved thanks to Beatrix Potter, author of Peter Rabbit, in there you can ride a steamboat and a steam locomotive (short rides), also rent and ride a slow motor boat or a row boat in the lake:
Edinburgh is also a good choice, with the best castle to visit (wherever possible, buy the "skip the line" tickets).
From there you can go to Fort William and ride The Jacobite, the steam locomotive used in the Harry Potter films:
In London, Liverpool, Dublin and Edinburgh, there's the option to buy a "city pass", that allows you to use a hop-on hop-off tourist bus and gives you free access to many touristic attractions, sometimes with "skip the line" included, it varies from city to city.
Keep in mind that flights to Italy, France, Spain, Portugal, Germany, the Netherlands and many other European destinations are very cheap and quick. They do tend to have a much harsher summer though, if you decide to expand your trip, I would recommend travelling during spring, even in London you may get temps above 86F/30C in the summer, on "the continent" it easily goes above 95F/35C.