Not easily.
There are two issues that make it complicated. The first is that this was a Court Martial, not a civilian trial. The Queen (on the advice of the Justice Secretary) can issue a Royal Pardon in the case of a civil trial, but while the Queen is Commander-in-Chief of all British Forces, the Army is under the command of Parliament (that's why it's not the Royal Army, unlike the Royal Navy, and Royal Air Force). For the Queen to issue a Royal Pardon over a Court Martial conviction there would have to be far more exceptional circumstances than normal, and the recommendation would have to come through the Defence Ministry, and the Defence Ministry is not set up with simple processes for that to happen. Regardless of this, while the Queen does have the power, it would be very unusual for it to be exercised until a case has gone through the Court of Appeal, and (possibly - I'm not sure if it has jurisdiction over courts martial) the Criminal Cases Review Commission. The 'Royal Prerogative of Mercy' is meant to be the final avenue of appeal is all else has failed.
The fact he plead guilty also complicates matters considerably. If he had gone to trial, there's a very good chance he would have escaped conviction, but he chose not to run the risk. That is his right, but it also limits his grounds for appeal.
To me, as an American, this whole case is repugnant in the extreme. The handgun at the heart of this matter is in no significant way different from the handgun which I carry, concealed on my person, on a daily basis. It is in no significant way different from the thousands of handguns sold legally over the counter every day in this Great Republic.
That Britain and the Commonwealth Countries see fit to treat these self-defense weapons as contraband, to be gathered up and destroyed by the government, the mere possession of which is ground for years of imprisonment is appalling. Your weapons laws (like ours ... indeed like any) serve only to harass the honest peaceable citizen.