Can you buy a screwdriver if you go to nearest town outside of London?

Try B&Q anywhere in London.  They should have a screwdriver.  I’ve seen them.

The tool or the drink?

No, you have to go to Inverness, speak Welsh with a Los Angeles accent and pay in gummy bears.

No.  People outside London only have crude wooden tools, and are afraid of iron, which they associate with witchcraft.

You can buy screwdrivers in London. It would be up to the police to decide if you were breaking any lockdown restrictions.

Yes; screwdrivers are widely available in and out of London. 

Vodka shortage in the UK?

Depends on the town and what shops it has.  At the very least, a local corner shop might have such a thing if there are no DIY stores in the area.