If I remember right, Pimoroni’s software tends to complain (and not install?) on operating systems other than RaspiOS, I take it because supporting other OSes becomes really complex when they do things different ways. So, the speed control won’t work unless you find a workaround for that.
However, the fan seems to spin when you attach it to the header, so it should run fine even without the software. The fan control doesn’t turn the fan on, it turns it off, so no fan control just means it runs at full speed.
The next question is pin conflicts: you don’t say which nanomesher HAT you’re using, but in theory it’s possible that the electronics on the Fan SHIM might upset whatever the NM HAT is trying to do with that pin, but we’d need to look at the exact pinout of the NM HAT to figure that out.