I have followed the same procedure with two machines recently, the most recent a Lenovo laptop preinstalled with Windows 10 Home x64. I removed the original HDD, fitted a new SSD, cloned the OS across and mothballed the original HDD until the warranty expires. The cloned OS failed to boot and could not be repaired so I refreshed the installation with the MS Media Creation Tool, no issues with activation at all. The second PC was also purchased recently but this was an end of line deal, this machine came with Windows 8.1 installed, this machine had room to accomadate an SSD alongside the existing HDD. I updated the OS on the HDD to Windows 10, cloned the OS across from the HDD to the SSD formatted the HDD as a storage partition and refreshed the SSD with the Media Creation Tool, again no activation problems. I was unaware that there was an issue with swapping hard drives, this is a new issue to me so I took no precations with regard the activation. Why didnt I just carry out fresh installations? Im a bit paranoid and getting more and more cautious these days.
It appears both computers were activated with OEM keys, If you want key numbers, see in MDL QUICK LINKS & FAQ's in Sig 28.) Get Product key OEM or Installed Key, Even will tell you which version of Windows it came from Here Or Microsoft PID Checker Here