As 'Clockwork Mouse' says, this won't go to court martial, but for such a public breach of discipline, punishment must be seen to have been carried out.
If the Guardsmen was caught dicking about solely by an NCO out of the public eye then I'm sure a bit of "wall to wall" counselling as you put it 'Brickhistory' would be adequate. Dicking about in front of one of the most photographed/filmed buildings in London, and allowing evidence of such actions to be broadcast around the world, is a slightly different matter.
This is a Guardsman who is a member of the Household Division on Ceremonial Duties. The very title Guardsman implies what he should be doing, not acting like a ****.