@Tomkitten,
What is the operating system on the new computer? Windows 7 does not use the same drivers as XP and HP has not been agressive in supporting older printers with new W7 drivers. Some ideas:
1) Look up the printer on HP's website to see if you have the right drivers. If you don't, install said drivers. If HP says you are out of luck, remember that next time you need to buy a product, I did.
2) Check the port the printer is using. If it is a USB printer, move it to another USB port and reboot. If you disconnect the printer, does the computer make a sound like something was disconnected?
3) Get a laptop somewhere and plug the printer into it. My guess is that it is fine, but it doesn't hurt to check.