No, his injuries don't seem to have affected his ability to command at all. Keep in mind that as an admiral, Nelson wasn't standing on the quarterdeck yelling orders to his crew; he had a captain with a full complement of officers to command his ship, and his role was to command the fleet, send orders to its captains, set overall strategy, confer with consuls and other political personages, and deal with enormous amounts of paperwork.

Nelson seems to have possessed a great deal of charisma that he could turn on or off at will, and realizing the limits of communication at sea, early on adopted a policy of communicating whenever possible during a campaign early and directly with his captains, often over meals or in a semi-informal setting. I wrote more about this style of command here and here.

There's an anecdote of Arthur Wellesley meeting Nelson before Trafalgar that I like to quote:

"Why," said the Duke, "I am not surprised at such instances, for Lord Nelson was, in different circumstances, two quite different men, as I myself can vouch, though I only saw him once in my life, and for, perhaps, an hour.

"It was soon after I returned from India. I went to the Colonial Office in Downing Street, and there I was shown into a little waiting-room on the right hand, where I found, also waiting to see the Secretary of State, a gentleman whom, from his likeness to his pictures and the loss of an arm, I immediately recognized as Lord Nelson.

"He could not know who I was, but he entered at once into conversation with me, if I can call it conversation, for it was almost all on his side, and all about himself, and in really a style so vain and so silly as to surprise and almost disgust me. I suppose something that I happened to say may have made him guess that I was somebody, and he went out of the room for a moment, I have no doubt to ask the office-keeper who I was, for when he came back he was altogether a different man, both in manner and matter. All that I had thought a charlatan style had vanished, and he talked of the state of this country and of the aspect and probabilities of affiars on the Continent with a good sense, and a knowledge of subjects both at home and abroad that surprised me equally and more agreeably than the first part of our interview had done; in fact, he talked like an officer and a stateman.

"The Secretary of State kept us long waiting, and certainly for the last half or three quarters of an hour I don't know that I ever had a conversation that interested me more. Now, if the Secretary of State had been punctual, and admitted Lord Nelson in the first quarter of an hour, I should have had the same impression of a light and trivial character that other people have had, but luckily I saw enough to be satisfied that he was really a very superior man; but certainly a more sudden and complete metamorphosis I never saw." source