Do you want to move your current Windows 7 HD to the other computer to see if it will run and reactivate? This is sometimes possible. Start it up and see it switches out all of the drivers
(requesting several restarts) and the performance is then good. Some have even reported that they didn't have to change out the activation key, but since the hardware has changed, it is best to reactivate at the Computer>Properties activation link to Change Product Key. Just reinsert the Product Key to see if it will activate - if not, make the robocall to MS which deactivates the key on old hardware and activates on new.
if you must clean reinstall to the new computer, you might need to do the workaround given here: Clean Install with a Upgrade Windows 7 Version
. You do not have to reinstall XP to reinstall Windows 7 as an Upgrade version, just use the workaround.
Any retail version which is not OEM can be moved to the computer of your choice whenever you wish; it just can't be on more than once computer at a time. This is why there is a robocall to MS: it deactivates the key on the old hardware and activates it on the new.
In your case, if you are fortunate to have the old HD with Windows 7 start up in the new computer, you have effectively reinstalled it on new hardware. You don't need to tell MS you moved the HD. I doubt you'll even have to talk with a person, just make the robocall. Again, others have said they didn't even need to reactivate but that is a bit hard to believe.
Finally, if you need a sure way to reset activation on an installed Windows 7, you run a Repair Install
. I have only had to do this once in about a dozen times of successfully moving HD's and changing the key at Computer>Properties link.