Assuming the PCs don't die quickly, they'll probably learn quickly that leaping into combat heedlessly can be deadly for all involved.
The healing rates in RMSS are very generous already, so I don't adjust them.
Without magic the PCs will pretty much have mundane healing (1st Aid, 2nd Aid, Surgery etc) and (depending on your scenario) herbs. Mastering either of these areas is not easy, but they can both be very powerful healing methods (herbs especially).
I have made my own spreadsheet detailing the possible uses of mundane healing skills, based on the rules found in various RMSS books (and expanded upon of course). With high skill levels there is much that can be done (eg a sucessful 2nd Aid roll could decrease healing times by 25% - not my rule, official..I think it is in SOHK).
The issues with herbs are that the details aren't all in one place (another little project of mine), although not as bad as for healing, the details for gathering and preparing them are sketchy (more home brew I'm afraid), and the price list is ridiculous (maybe it could be used to describe the relative values of herbs one to another, but if those numbers are applied as is then every peasant in the land would be out looking for herbs - finding just one of moderate priced herbs would be like winning the lottery).
If you have the other options mapped out, then the loss of magic doesn't have to be quite so devastating. Of course if you are playing an accurate medieval european setting then medicine will be rudimentary at best (even Richard I died of gangrene, and he was a king).