Well, cauterization is actually medicine. For a non-medical description, cauterization is just cooking the meat until it stops leaking. As far as I know this would the main treatment for traumatically losing a limb in a pre-industrial setting.
I’m not sure if the characters control over fire would be precise enough to do the cooking personally, but it’s certainly an option, even without resorting to magic.
This is of course assuming the characters have the medical knowledge to realize that cauterization is a legitimate solution to the injury. Which is certainly plausible.
That said, in a high fantasy setting, with advanced healing magic, it’s possible that cauterization would be viewed as repugnant, or a perversion of magic. Obviously, that’s getting into the ethics and cultural norms of your setting, but it’s probably something to think about.
If it is something they’d avoid, then an impromptu tourniquet would buy a little time, but getting enough pressure to actually stop an arterial bleed out might not be something your characters are capable of.