You can use Anytime Upgrade to easily unlock the extra features in Ultimate from the Basic desktop.
Just type Anytime in Start Search box, insert the Ultimate key.
You should have your Ultimate Product key backed up like a bank account number. If you cannot find it in your emails, then run Belarc Advisor
to audit all of the installed hardware, software and Product Keys before deleting Ulimate from the old machine.
But before activating Ultimate on the new machine I'd delete Ultimate from the old one by booting the Windows 7 Basic installer, choose Custom Install, then use Drive Options to Delete the OS partition. You can then reinstall Basic to the machine using the Product key on COA sticker on machine following these steps to get a perfect reinstall: Reinstalling Windows 7
There is also a workaround to change to a lesser version of Windows 7 doing an in-place Upgrade. Let me know if you need it.