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&quot; 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&quot; 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&quot; fork has 1&quot; = 25.4mm outer diameter. The inner diameter is 22.2mm
A 1+1/8&quot; 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&quot; fork.