I found several references that may be of interest. The gold knife/fork combination appears to be a gift from Lady Spencer, as detailed in the first two references:It was probably at this time of her first intercourse with Nelson, when he had just lost his arm, that the enthusiastic Lady Spencer presented the Admiral with the combined gold knife and fork, manageable with one hand, which still exists.
Nelson in England: a Domestic Chronicle by Esther Meynell - 1913The Lady at the Admiralty was Lady Spencer, formerly the lovely Miss Lavinia Bingham, eldest daughter of Lord Lucan, whom Gibbon described to Lord Sheffield after seeing her at Lausanne : " She is a charming woman, who with sense and spirit has the playfulness and simplicity of a child." The Naval Commanders who frequented the Admiralty during her husband's administration felt the influence of her superior mind ; they often touchingly referred to her in their letters as of one who would sympathise with them in their difficulties. She was all to them in England that Lady Hamilton was in Naples, and the gallant sea-officers valued and respected both women. Lady Spencer had the penetration to appreciate the genius of Nelson through the element of personal vanity which frequently enveloped him. When he lost his arm she presented him with a combined gold knife and fork.
Nelson's friendships ... By Hilda Gamlin - 1899
The same piece appeared at the Nelson Centenary in 1905The cocked hat worn by Nelson at Copenhagon was given to the museum only a couple of years ago by Messrs. Widdowson and Veale of the Strand. It had been in their possession for many years and had figured prominently amongst silver statues and trophies. Mr. J. A. Mullens has been a liberal donor of Nelson relics both to this museum and to Greenwich Hospital, and the pair of ice-pails purchased at the 1895 sale at Christie's are particularly beautiful works of art. At the same time he also gave the gold combined knife and fork which Nelson used after the loss of his arm.
The Nelson Centenary Exhibition at the Royal United Service Museum, Whitehall. - 1905
And it would appear from the reference below that these pieces were commonly known as 'Nelson Knives':Suggestions are made for improvements in Government limbs supplied to the army and navy ; and these are, mainly, that the wooden-pin leg should have a catch-joint for the knee to enable the wearer to sit comfortably, and in the case of the upper limb the useless unjointed stump of the whole arm should be discarded, and the 'stump arm below elbow' should have an artificial wrist-joint as well as the longer stump. Both should have at least three or four useful instruments given with them, and one of the instruments should be a Nelson knife, which is a combined knife and fork.
The London medical record, Volume 14 edited by Ernest Abraham Hart - 1886
Perhaps another one of Nelson's forks was exhibited at the Naval Exhibition in 1895:
And take a close look at this image from 1895.