Can the Queen or the Prime Minister issue a pardon?
If so, why hasn’t it been issued already?
This SAS serviceman is an exemplary model of what should be good for Britain.
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.