Who owns cultural heritage in the UK?

Answer 1

Ownership is split between various bodies.
Museums and Art Galleries can be run by the Government, local authorities or private trusts.
Many ancient monuments are run by English Heritage or the equivalent body elsewhere in UK.
Cathedrals and Church's are owned and run by the Anglican Church.
Lots of Stately Homes are still owned by the families who built them.
The National Trust owns hundreds of properties and huge tracts of land, particularly around our coasts.
Most of the land in our designated National Parks remains in private hands, but is strictly regulated.

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Answer 2

Every last citizen of the United Kingdom.

There's a very nice answer immediately below mine!