Hampstead in London is well known for big expensive houses inhabited by millionaires. The Bishop's Avenue there has been called Millionaires Row. Mayfair and Kensington are expensive too though you're unlikely to have a big house there - no space in the middle of London! Actors and other rich showbiz people tend to frequent west Surrey, especially if they play golf - the place is full of golf courses.
Denham is another slightly showbizzy place just west of London because though it is a small village, it used to have a film studios and I know of an actor and a singer who live there. (Though not everyone there is rich - my mother lives there too!) Loudwater outside Rickmansworth, a bit north of that, has the highest house prices in the UK though I don't know of anyone famous there.
Going further north, Alderley Edge in Cheshire is popular with rich footballers, mainly from Manchester United.
Hampstead
Cheshire
Kensington
Cotswolds
In the UK, most really rich people tend to prefer a quiet lifestyle, and prefer not to have celebrities, etc, moving into near them attracting reporters, fans, etc, with them. Places where the really wealthy live are often out of the way and get very little media attention (as the owners of the media live there and tell their staff to leave the place alone). Hence the places where pop singers, footballers, etc live are usually not the same ones where the financiers and industrialists live.
Holland Park west London,
Prestbury in Cheshire
Sandbanks in Dorset,
Totteridge north London,
Cheshire
Weybridge (Surrey)
Esher (Surrey)
Belgravia (London)
Kensington (London)
Chelsea (London)
Mayfair (London)
St Johns Wood (London)
Notting Hill/Holland Park (London)
The Cotswolds (Gloucestershire)
Sandbanks
Cheshire
Wiltshire
Oxford
London (of course)
Several others.
Belgravia in London. Around Sloane Square. The "Sloane Rangers" are the idle rich young girls who shop there. Prices are rather high!
Belgravia, without a doubt.
A lot of rich people in Surrey.