Wolfram don't provide support with Home Edition and suggested that I ask on this forum.
I hope it is because we are so awesome and not because we don't have to be paid.
It refused to license itself claiming that it had been installed on a 'different' computer.-- mac addresses are still the same.
The MAC address is not the crucial thing here, because a desktop license is always bound to the machine and the operating system and Wolfram uses several things to determine whether the environment you installed Mathematica has been altered. Therefore, a change of the Linux distribution invalidates your license.
I'm not sure whether I can help you, but usually when you bought a Mathematica license (and probably you have to register it) and you created a login at https://user.wolfram.com you should see your product when you navigate to My Products and Services:
When you click on the product you bought (in the image above it would be the Mathematica link), you get the details to your license and nowadays you can manage your Activation Keys from there. In this page, you have the opportunity to deactivate an activation key which is currently in use
After deactivation, you can reuse the activation key (which is displayed in the first column of this table, which is not visible for the obvious reason).
If the situation for a Home License is completely different, then you could have a look in the Mathematica knowledge base which has a separate section about activation and license problems. Unfortunately, most articles end with please contact the Wolfram support).