In theory, any "700C" tire will fit that rim (including so-called 29" mountain bike tires), but beyond a certain width the tire tends to wobble side-to-side if there's not enough pressure or spread the rim's sidewalls apart if there's too much pressure.

In my experience, a rim that originally came with 700x23's will work fine with 700x28 tires, could work with 700x32's, and is getting iffy with 700x35's.

(Are you sure those are originally from a 1981 Raleigh Competition? Even racy bikes of that era rarely came with anything narrower than 700x25's.)