England, currently dividing my time between university in Guildford and home just outside Reading
Too many to list here!
Partly correct. Your stem quill is 7/8", which is 22.2 mm. The steerer tube of your fork is 25.4 mm, which is 1".
There may be another problem though. I seem to recall some Raleigh Sports using a non-standard clamp size for the handlebars, 15/16" (23.8 mm) rather than the more common 1". This won't affect the stem size, but you may need to replace the handlebars as well, as the 23.8 mm clamp hasn't been used on modern bikes for quite a while, meaning you almost certainly won't find an adjustable stem in that size (I doubt they were ever made). Do you have a set of calipers to measure the diameter of the clamp area on the handlebars?
You may find the length of the bulge is smaller the length of the stem clamp. I am not talking diameter, I mean the length of the section of bar that is bulged. If so, you may not be able to get the old bars to stay put in the clamp. You may need to find a new handlebar to match the stem.