If you are visiting the London area / South East, someone's already mentioned Arundel and Leeds Castles (you can visit both by train - a car isn't necessary for these two). You could add to this the wonderful Bodiam Castle (easiest by car) in the middle of a lake. I just looked up the web sight http://www.sussexcastles.com/castles/
and it shows 11 castles in the twin counties (East and West Sussex) alone. Kent has many as well including the best known at Dover, Rochester, Leeds you know about already etc. Going north, not too far from London are Framlingham and Orford castles, both largely intact. West - don't forget Windsor, and in London of course there is the Tower.
Further afield, someone's already mentioned Wales, which has perhaps the largest and most complete castles, mostly built by the English to overlord the rebellious Welsh. Scotland has mainly smaller examples, built in a completely different style.
Essentially, the UK consists of 4 nations (in the case of Ireland, just a part of one), with England dominant because of its size, wealth and history of conquest. In recent years, a limited devolution has taken place, with Scotland and Wales having parliaments, which are still subordinate to the union parliament at Westminster. The subject is huge, with perhaps as many opinions as there are Brits! Most of our large cities are now highly multicultural and diverse. Some of these localities are also highly photogenic, and in London there are several Hindu and Sikh temples, some of which welcome visitors and afford excellent and photo opportunities.
I live in Brighton - the nearest castle is Bramber which has about 2 walls left but is great for picnics. You can get there on a city bus from Brighton (2a, I think, destination Steyning). Also Lewes Castle - Lewes itself is great to wander round and there are nice shops and cafes. There is lots to photograph in Brighton and Hove city including the Pavilion, the colourful Lanes, the North Laine and of course the sea front and pier.
Lens-wise I would take at least one wide-angle. I make do with the 17-40 along with my 24-105 and 70-300 L. As others have said, allow for the weather being variable! If you are going in summer, a light-weight waterproof is advisable, but we do have shops here if you find yourself without! Have a great visit whatever you decide on.
PS I have just seen Tom Scott's images. Beautiful!
On further thoughts, you don't say what time of year you are coming. Spring and autumn are probably the best, but summer and winter have their virtues. I assume you are coming from the US. We are actually a bit further north than pretty much all of the United States (except Alaska) being beyond the 50th parallel, so expect very long days in summer, especially in Scotland; conversely very short in winter. Tom Scott's image are taken in winter (well obviously with snow). In Cumbria (nearly Scotland!) the golden hour in winter would be most of the day. Even here in the South, the sun rises before 5am in June and sunset at around 9.30pm, and in winter daylight is strictly 8 till 4. By the way, feel free to get in touch if you are nearby firstname.lastname@example.org
and if it's practical I could show you a few places.