Original fork 1'', new fork 1 1/8''

Your frame seems to have a non-standard head tube diameter or you mis-measured it.

Here you can see a list of headset standards: https://www.parktool.com/blog/repair-help/headset-standards

There is no headset standard where head tube internal diameter would be 31mm.

For 1+1/8" forks and bicycles with external headset cups, the inner diameter of head tube should be 33.8-33.9 mm.

Only if you can find a headset that is compatible both with your frame and with the 1+1/8" fork, you can install it.

I would measure the head tube internal diameter again. The external pressed-in headset cups might be affecting your measurement.

Do note that 1" fork has 1" = 25.4mm outer diameter. The inner diameter is 22.2mm

A 1+1/8" fork has 28.6mm outer diameter and 25.4mm inner diameter.

So if your fork had 25.4mm inner diameter, it's a 1+1/8" fork.