Hi
First make sure that you have logged into Window 10 on the computer using your Microsoft account.
Theoretically once you have done that your are allowed to reinstall as often as you want, it does't ask for a registration number anymore.
After that all I can tell you is that I switched from a conventional HD to an SSD and all I did was to install Windows 10 on my new drive, and sign in with my Windows ID.
It was somewhat different for me because I just added the SSD to my already installed hard drives, but I did have only the SSD connected when I did the installation.
It didn't question my registration, just logged in as normal, and then I connected my other drives.
But it you have problems you will still have your original...