Oh yes, Martin, well done.
The pitting towards the handle looks worse in the full-sized picture than the smaller patch below and to the left of the name stamp...but cannot tell. I doubt it is a historically important saw--that'd be one for Alf or BB, what me having no books and being too lazy to do any web research.
But if it were mine, I would gently scrub it with a citric solution using a scrub pad just until most of the rust was gone and the steel an even patina. I probably wouldn't go over board and make it bright and shiney.
The handle I would scrub with some white spirits, again until the gunk was off, but not til the wood was like new. And be easy around the saw nuts. If they get too shiney compared to the remainder of the handle they'll look out of place.
Looks to be about 2 1/4" from teeth to the bottom of the back or so? At that length I would imagine it was a tenon saw, now sharpened to the point there's not much life in her for that. But it's a nice looking saw nonetheless. But then, I have a thing for them.
I probably wouldn't use the saw personally. By the time the teeth were jointed and refiled there would be that much less life. If it's an old enough saw, one runs the risk of snaping teeth off just sharpening it. If you try, start back by the heel and go gently.
So if it were mine (and it wasn't historically significant) or it was sent to me to change (been done), and the steel was in good enough shape, I personally would cut it down and make a DT saw from it. Maybe 8"-9" long. A more sprung, open handle. But then, there are people who would slap me about the face for even suggesting it.
Take care, Mike