Looking around for a config or setting file for putty to copy the settings onto another machine, I discovered they’re actually in the registry.
Luckily they’re all under one root.
Just run this export command to create a file you can copy to a new machine:
regedit /e C:\puttyconfig.reg HKEY_CURRENT_USER\Software\SimonTatham
Copy to new machine.
Right click it.
That’s it, your done!