I would bet that disposable scalpels followed closely behind disposable razor blades for shaving because they are very similar products and could be manufactured together. I'm going to guess that was in the 1950's.
As far as non-disposable scalpels go, they have been in use for a very long time. In the Civil War era, they were most likely sharpened by hand the same way razors were sharpened, on something like an Arkansas stone if necessary and more often just touched up on a leather strop. I don't think they did a lot of sterilizing back then either.
Between then and the advent of disposable scalpels, they probably used whatever the prevailing best technology was for sharpening, but it didn't matter much what method was used because the scalpels would have been cleaned and sterilized after sharpening anyway. I would assume that at some point there were "medical grade" sharpening devices that cost 5x their identical industrial counterparts.