No one has battleships anymore. Only the USN has a few decommissioned Iowa class battleships.
The USN does have the Ticonderoga class which is a cruiser, but the cruiser designation doesn't quite imply greater combat power.
Unlike USN destroyers, British destroyers are very specialized towards anti-air defence and have very little anti-ship or anti-submarine capability. Currently, their only anti-shipping armament is the 4.5 inch gun on the bow and the missiles carried by its helicopter.
To compare, the USN has around about 83 active guided missile cruisers/destroyers. The RN has 6.