The academic answer is 'No Change', but the correct answer is 'probably will be different'. How much will depend most on the replacement tire. Quite a few factors affect the actual operating scrub radius besides geometric settings. Run-Flat, unidirectional, brand, pressure, rim width, tread depth, raised letters, odd or even number of belts, and radial vs. bias produce consistently measurable differences in tire Mx (Overturning moment) properties. You need access to raw tire data to observe these tendencies because just a stiffness or a fitting model supplied by a vendor can/will obscure the operating functions of the tire at your loads, alignments and wheels (yes, the test wheel can become a player because its usually chosen to minimise wheel effects for tire validation reasons and to eliminate the wheel as a source of lateral stiffness loss).
If you have some air bearings or grease plates, you can sometimes measure a scrub radius CHANGE under lab, non-rolling conditions, especially if its pretty large (25+ mm) already. You need to determine the arc center of the plate movment and compare it to the center of the tire position. In reality, the center of the tire is NOT the centroid of all the forces and moments, but you can get the hang of it. Since the wheel is wider, the tire spring rate is usually different, so you may have lost the 'same diameter' constraint.
You probably won't read any of this in a book. Get someone to invite you into a laboratory tire test session and have them crank it to the load (Fz), slip and camber angles that your car runs and observe the Mx/Fz math channel.
I can probably come up with some sample results to post if I can figure out how to show pictures/graphs AND after I get my barn's burst water line fixed. Stay tuned...
