I recently bought a new pc, and was a bit quick to install Windows, create a backup drive and remove the disk to install the larger ssd from my previous computer.
The upshot of that was that, after installing the operating system on the new drive, it wouldn't activate Windows as I didn't fully activate it on the first install before making the recovery drive.
In the end, I got in contact with a Windows support person who recovered the activation key from the computer hardware. Which is pretty simple when you know the commands.
To get your key:
Right click on the start menu and select "Command prompt (admin)"
Type in:
wmic path SoftwareLicensingService get OA3xOriginalProductKey
You should then get your key in the console output.