All of the classic "Age of Sail" era was pre-Lister/Simmelweis so there was virtually no sterilization and any disinfectant effect from a poultice or the like would have been accidental and unique to the surgeon compounding the salve. Alcohol was used, when possible, to sedate patients before, during, and after procedures. If instruments were rinsed or cleaned between patients, it would only have been done to suit the fastidiousness of the surgeon in question. There were no requirements for them to wash hands or instruments between patients. Moreover, the sick and recovering where generally kept on the orlop deck below the waterline. This was for safety during battle, but in practice this was the sickbay as well....in the least well ventilated portion of the ship with almost no fresh air and quite near the perpetually filthy bilges just below. Some captains/surgeons would bring the sick on deck for "better air" when possible, but this was not a required practice.
When not in a rush, the surgical instruments were heated in a flame to make them less of a "shock" when cutting a patient (this would have had a limited anti-bacterial effect as well but would have been a by-product, not intent). They WERE aware that foreign objects internally almost always resulted in infection (though they did not know why) so they did their best to remove splinters and cloth from wounds (though this was imperfect).
Interestingly, that was one of the reasons officers and others wore expensive silk shirts in battle when possible. Silk's tensile strength is such that, when hit by a relatively low speed penetrator (black powder musket ball or the like, particularly from beyond point blank range) the silk cloth often held together and came out of the wound entirely when the shirt was removed or, in some cases, actually surrounded the musket/pistol ball and could be used to pop the ball out of the wound entire. The same was true to a lesser extend with thrusting wounds from swords and the like -- silk made it less likely for cloth to remain in a wound. So the dueling in silk shirts thing you see in films is actually correct (though the producers seldom had that in mind during the filming).
There is an amusing scene in one of the later O'Brian novels that touches on this ignorance of modern microbiology. Dr. Maturin and a colleague stop their scientific dissection of a dolphin corpse to have their main meal. The simply take and use the dissection knife to carve the bird they then eat.
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