Was it survivable? Sure. The rates of survival undoubtedly varied greatly from culture to culture, however, but this sort of injury is not a death sentence.

When you amputate a finger, hand, or arm the blood loss is actually often not that severe, even though you'd think that by severing an artery you'd end up with a gusher that would result in quick death. The human body has some built-in mechanisms to deal with traumatic amputations (which can happen in the wild) that increase the chances of survival. If you cleanly cut (transect) an artery it will usually constrict and retract into the body, which significantly limits blood loss. The other side of the coin is infection, which is a crap shoot with a big open wound, but not inherently a death sentence. Additionally, quite often when an amputation was performed it was followed up by cauterizing the wound with heat.

The best case scenario for a victim would be to have the amputation performed in one clean cut by a very sharp sword or ax and then to have the wound be cauterized immediately. This wouldn't guarantee survival, but many more would survive it than not.

We don't know all of the methods that every historical culture used in performing amputations, of course. We know that some practiced cautery, for example. We do know that punitive amputations seemingly have a history going back several thousands of years, which indicates that it was generally significantly survivable.