One of my PCs has Tinkerer and spent 1 rank on another character's Vibro-axe

In the session coming up, the player wants to use another character that does not have Tinkerer.

Will the effect of Tinkerer stay in place? Or not?

Is the player permanently switching out characters, or just trying something else for a run? If it's the later, I wouldn't worry about it, and let the tinkering work fine. If it's the former, I'd have the Vibro-axe return to normal, or if you're feeling more generous, break on the next despair.

If the character with the Tinkerer talent choses to play another character, I'd rule that the extra hardpoint gained by Tinkerer disappears. Unless, of course, the player who plays the character with the vibro-axe is also playing another character; then it doesn't really matter. Otherwise you could very well end up with a character who takes all the ranks in Tinkerer, fixes up everyone's equipment, and then just gets parked somewhere while the player uses another character. That's not really how it's supposed to work. If you chose to give another player the benefit of the Tinkerer talent, my opinion is that you should play the character and accept the consequences of your decision - i.e., that you have paid XP for a talent someone else is gaining the benefits of.