I was born in London and now live in Birmingham so can be fairly unbiased.
London is a major world city, along with Paris, Madrid, New York etc. It can compare with any of those cities for all the things you listed.
In my opinion the UK has no other "world" cities.
While Manchester, Birmingham, Newcastle, Bristol etc do have their merits, none of them can compete with London for its full range of facilities.
For example Birmingham has about 3 or 4 theatres in the centre, London has about 30.
EVERY major play that opens always opens in London, and the London cast are always the top actors. Once a play goes on tour round the UK you have the second or third string actors.
London is the centre for government (Parliament, Downing St), Royalty (Buckingham Palace etc), Shopping (Oxford St, Bond St, etc), the Arts (loads of major galleries).
It bet almost every chain store in the UK (particularly womens clothes shops) has its largest store in London.
It is also the centre for food (with thousands of places to eat), and the cente for media (Newspapers and other media) also the centre for music, comedy.
I think your last sentence sums it up - London is the best place to be from a foreign point of view (I bet it gets more foreign tourists than all the other UK cities put together).