The beach may be a prohibited area due to the number of people on it.
As such that would, as I understand things, be in accordance with CAA rules regarding flights over people/crowds although it need not be a defined exclusion zone. There is however something in the back of my mind about beaches being explicitly mentioned in the CAA 'literature'.
That said the land owner has the right to decide that drones can not be launched and or landed from their land. That probably extends to the land owner having the right to refuse permission for the pilot to be on their land whilst flying a drone.
That said, the tidal zone is, I believe, normally "Crown Land" and I recollect that, in general, the Crown permits drones to be flown from the tidal zone though I seem to remember reading of some excluded areas.
As far as I know the CAA govern the airspace but personally I would question the wisdom of flying over a crowded beach even if it is legal to do so. Why, what happens if something goes wrong and the drone comes down on the beach?
There's a fair chance there are quite a few "accustomed to humans" gulls knocking about, on the hunt for edible scraps, and the drone is likely to be in their airspace, not a particularly good combination as far as I am concerned.