Hi there, I answered a very similar question awhile back, which may be of interest to you. The short version of it is that injuries of the sort Nelson suffered (an amputated arm and blinding in one eye) were neither particularly unusual, nor considered to be disabling, in the context of the British officer corps. (Commissioned) officers were not expected or meant to be racing up the rigging of a ship or otherwise engaging in major physical activity; warrant officers might be expected to be more physically engaged in their jobs, and a one-armed seaman would be unusual but not unheard of. But for an officer, partial disability was not a bar to command. Nelson was already a post-captain when he lost sight in his eye, and his arm wound came later; given the promotion practices of the late 18th century/early 19th century British navy, once a man were made post he would die an admiral, if he lived long enough -- promotion was based strictly on seniority (though commissions were decided by the Admiralty and its chain of command, and captains could become "admirals without distinction of squadron.").

I've written about Nelson's individual characteristics before, which you can find here; the relative youth of officers here; the British chain of command here; and Nelson's tactics at Trafalgar here if those are also of interest. Let me know if you have follow up questions.