Can I migrate Windows 10 system disk to a different OEM licence?  

Your old drive was probably set as legacy and your new is trying to boot as EFI.

Please give us the specs of your old an new computer.

OK so I managed to resolve the issue by booting to a Windows 10 Pro disk and using the repair option. Windows started up but, as predicted, is not activated. The licence status says: "We can't activate Windows on this device because you don't have a valid digital licence or key., select Troubleshoot below. Error code: 0xC004C003

So I ran the troubleshooter, and found the Windows licence for the machine, but when I tried to activate it I got this message:
"Unable to activate Windows. We can't reactivate Windows on this device. Try again later".

I haven't used the new Windows licence yet, and SLI -dli tells me the old licence that I had was RETAIL (not OEM as I thought). So (how) can I reactivate it on the new motherboard?

Try:
slmgr /ato

NavyLCDR said:

Try:
slmgr /ato

Hmmm...when I tried that I got: "Error: 0xC004C003 The activation server determined the specified product key is blocked"

???

Hmm. Is the old drive MBR and the new computer set up to expect GPT? If so, maybe it's just a matter of enabling CSM in BIOS.

No, probably not. Then you just wouldn't find a bootable drive.

Did the new computer came with same version (Home or Pro) that you had on the old one?