Information is sometimes hard to find because of security issues, but I believe that a number of Crowned Heads do keep houses in the UK. These are not just London bases [although these are popular] but country houses within fairly easy reach of London: in Surrey and Berkshire, to name two popular counties. I think that some Crowned Heads also keep a place in Scotland.
Many royal and princely families without their own accommodation in the UK tend to stay at Claridges. Princess Grace used to stay in the Connaught. This latter hotel is very grand indeed - it has (sadly) been re-vamped quite recently and I feel that its modernisation resulted in it losing its almost 'timeless' feeling of grandeur and luxury, which has begun to be replaced by this sort of 'modern luxury that is not really real luxury' feeling, which is too hard to define. Some of my family always stay/ed at the Connaught - it is slightly more 'tucked away' and therefore more private than some the Ritz and Claridges etc.
I do have a very funny story about Princess Grace. Years and years ago, I spotted a very young [teenage] Princess Caroline outside The Connaught. She seemed to be having a row with the Doorman. There was some fairly audible shouting. One of my aunts happened to be staying at the hotel at the time, and I had been visiting her. She later found out what had happened. Apparently Caroline had turned up to visit Princess Grace, who was also staying there at the same time as my aunt. [possibly for the same reason - a flower arranging symposium was on in London at that time.] The Connaught had [and I think it still has] a very strict 'no jeans' policy. Princess Caroline had turned up in jeans. The Connaught Doorman [in accordance with this policy] had therefore stopped Caroline from entering. Caroline was very angry and so apparently sent a message to Princess Grace, complaining that the Doorman would not let her in becuase she was wearing jeans. Whereupon Princess Grace had sent back a message: 'Quite right too! First apologise to the Doorman, and then go and change your jeans and find something else to wear'. I always think that this story presents Princess Grace in a very good light - a mother insisting on proper behaviour and not trying to 'pull rank'.
Hope this amuses some of you,