I'm no expert, but the lower spool should be readily removable - held
by a couple of clips or something - so it is easy to change colours etc.
I don't think it should require a screw to be removed ! It' suggests to
me that it may not quite be in the right position, which *may* explain
why the lower thread is not always being caught. Apart from that, it
sounds as if you're doing everything about right.

Harry Bloomfield wrote:
Very old 1950's basic Singer sewing machine (201K), which was given to
us twenty odd years ago and we never needed it until now. It is one of
those black painted things with gold decoration and the optional
electric motor.

The instructions are still with it, so I'm fairly sure it is threaded up
correctly and it looks right, but it doesn't sew. The lower spool of
cotton just sits in the bottom, with a large very tight screw head below
it, which I have not been able to budge.

From the beginning...

The upper thread goes down through the hole with the needle, is dragged
around by the hooked rotating thingummy underneath going around the
lower spool, it brings the lower thread up as it should [1]. Leaving the
two cotton threads out the back I insert the cloth, bring the foot down
and start to stitch -except it doesn't.

It doesn't actually create any stitch in the material, it just moves the
material forward with no stitches, then eventually the thread knots,
jams and then breaks.

[1] The hook doesn't always catch the thread, when trying to get it set
up to start sewing.