One of life's mysteries...
The term "convention" springs to mind. The same question as to why captains generally sit on the left in multi-seat aeroplanes. Having come home to Oz where everything is strictly regulated, I have searched extensively on the latter to find that an aircraft captain can clearly sit wherever he likes because there is no rule that says anything else.
I guess the same applies to pilots mounting? They can choose which side is most convenient.
Other than that's where the ladder fits (making a right hand mounting most convenient), I am not sure why the SHAR and other RN jets came up the right hand side. Given aeroplanes were usually ranged along the starboard side of the ship, one reason could be that when manning up with a big wind over the deck it felt better to be forced onto the ladder and into the cockpit rather than the other way around.
I suppose that's also a possible reason why, when we had all the crab jets on board as well, they often used to be ranged along the port-side?