Just an update to this to give a bit more info on where the issue may lie, having had he same problem again on a new laptop.
If I went into HX Edit, it had the option to authorize computer, but it didn't do anything when clicked, and this is all despite the license manager saying the computer *was* authorized. I was signed in but HX Edit thought it wasn't authorized.
I went into the license manager and deauthorized it so I could go into HX edit to authorize it, and only then did the option now say 'Deauthorize Computer' without changing anything else, so it was now automatically authorized and Native now works, and my computer still shows as active in my active devices on the site. It appears the bug is in the license manager somewhere.
TL:DR
Deathorize if authorized using the license manager.
Sign in using HX Edit and let that authorize the computer.