I have just setup a new home PC. Windows 7 Pro. When I try to setup FireFox sync, I go to the 'I don't have the device with me screen' and fill in my account details and recovery key, but when I click 'Next' nothing happens. Once the username, password and recovery key is filled in, the button becomes live, but clicking it does not do anything.
I have no extensions installed but have also tried the process in safe mode and the same thing happens.
Uninstall your current Firefox version, if possible, to cleanup the Windows registry and settings in security software.
Do NOT remove personal data when you uninstall your current Firefox version, because all profile folders will be removed and you will also lose your personal data like bookmarks and passwords from profiles of other Firefox versions.
Remove the Firefox program folder before installing that newly downloaded copy of the Firefox installer.
It is important to delete the Firefox program folder to remove all the files and make sure that there are no problems with files that were leftover after uninstalling.
Your bookmarks and other profile data are stored in the Firefox Profile Folder and won't be affected by an uninstall and (re)install, but make sure that "remove personal data" is NOT selected when you uninstall Firefox.
If you keep having problems then also create a new profile or reset Firefox.