I would recommend a visit to St.Thomas Operating Theatre in Southwark, in London, if you can manage it. It is the only surviving original nineteenth century operating theatre extant http://www.thegarret.org.uk/. They didn't just walk around with bloody hands, they did usually clean them, as well as wiping the blood and pus of their instruments. However, until Pasteur proved the existence of germs in the 1860's, surgeons were unaware that infection was caused by invisible bacteria. So, although would wash their hands if obvious blood, pus or whatever was on them, they wouldn't have been aware of the necessity to wash hands in carbolic soap and very hot water, for example, to kill the invisible bacteria.