As early as 1834, the expression loose screw was used to illustrate a flaw in state affairs. In Martin Chuzzlewit, Charles Dickens wrote “I see well enough there’s a screw loose in your affairs”.
But the word was used well before that. For instance, in 1810, Sporting Magazine said “the others had got a screw loose”. I don’t know what the magazine meant by that.
But the same magazine in 1822 said “a screw, it seems, has been loose between Neat and the champions of England”. I don’t know what that meant either and I suspect the writer wasn’t too sure either, but the writer was suggesting something was wrong.
But I think the early meaning was implying something was wrong.
My opinion is that as the years progressed the meaning of “something wrong” gave way gradually to something wrong in a person’s head, without being too critical.
Now, whenever we say Bill has a screw loose, we mean we don’t agree with his line of thinking. Of course, Bill might be correct.
The expression isn’t meant to be too condemning. It says about the person it is aimed at that he or she is eccentric or a bit neurotic, without going too far.
There are many expressions that include screw. My big dictionary in eight pages says of screw: “The general name for that kind of mechanical appliance in which the operative portion is a helical groove or ridge (or two or more parallel helical grooves or ridges) cut either on the exterior surface of a cylinder (male screw) or on the interior surface of a cylindrical cavity (female screw), hence applied to various other contrivances resembling this”.
In the very early days, screw came from the Latin to mean a female pig, possibly because of the corkscrew tail, and has had a few meanings since then, some unsavoury.
The word screw has many meanings, applied to billiards, cricket and rowing and a few slang words in Australia. This country only seems to have the slang words.
If you want some other words to explain your friends, you could try a sandwich short of a picnic, bats in the belfry, cockeyed, crackpot, out in the sun too long, round the bend and a few other words I could list but decline to do so in case they get me into trouble.
But be careful how you use them. Remember that some people don’t have a sense of humour.