The brain can survive - but will increasingly fail - for about three minutes but I imagine you'd go into shock as the blade touched. I haven't tried it!
Execution in a decompression chamber in which the air is progressively pumped out must be quite a humane, almost pleasant way to go. It's rather like becoming slowly drunk and, yes, I have tried in under medical supervision and for a valid reason.This is not the same as asphyxiation when carbon dioxide builds up in the body causing increasing distress, which must be a horrible death.
Lethal injection to me seems almost the nastiest of the lot: there's something particularly distasteful about dressing up something brutal and savage as something clinical. Who inserts the needle; who presses the plunger? A least the guy with the axe is a honest thug! And it can be slow and painful.