trying to recover product key

I bought new windows product key , and windows 10 works great but my computer is to old for windows 11 , so I upgraded my hardware but when I went to update windows Microsoft could not find my new product key because it had been attach to my old mother board and not to my account ; I still have the old board so I'm wondering if I get the old system going would I be able to recover the product key , and does both of the computers have to be on in order for Microsoft to transfer the key to my new computer.