"Birmingham" or "Manchester" screwdriver?
Solution 1:
On this side of the pond the phrase "Yankee Screwdriver" is not uncommon. I also understand that during WWII the Dutch used it (derogatorily I suppose) after seeing American GIs pounding screws with hammers.
As it turns out, you CAN drive a screw part way with a hammer (coarse tooth deck screw, Robertson head, in softwood only of course...), saving a lot of time with little detriment on its holding ability*. In this sense the term is not necessarily (or strongly) derogatory.
*But in this era of electric screw drivers this is "old technology".
Solution 2:
I have only heard the terms "Paisley Screwdriver" or "Glasgow Screwdriver" used in the same way by natives of the opposing places.