Former French executioners reckoned beheading even with a guillotine amputates the head instantly, but it does not die instantly.
Even if it did, it's still a revoltingly gruesome and barbaric method.
I don't know if being hanged by gallows (not "hung", nor by gibbet) is any quicker, though it's said to be instant.
The air bubble in the arm is not guaranteed to do anything other than induce an appalling attack of "the bends".