Does the UK exercise rights over the resources of its continental shelf?

Yes. 

The Continental Shelf Act 1964 gave the UK Government rights over the continental shelf and led to the division of the shelf, for licensing and exploitation purposes, into quadrants and blocks. The boundaries of the shelf, originally set in the Continental Shelf (Designation of Areas) Order 1964, have been extended since 1964 by several Continental Shelf (Designation of Additional Areas) Orders.