I think with the Dell disk you're ok (but you cannot use the same registration key from a Dell on another PC) only because the Windows disk is just that, a Windows disk. Dell doesnt add anything proprietary to it including drivers, which means it SHOULD install.
Now I am not 100% sure on that, especially the legality of it, but I would think it would be ok to use if you have your key from the HP or Gateway. Microsoft would be the correct place to ask that question. Now like Phil said, the HP registration key might not work after all it is a Dell OEM disk.
You're almost safer calling HP or Gateway and spending the $20 to $40 to get a new disk (at the very least you'd probably get a driver disk as well) or buying a new full registration Windows disk.
Microsoft is very particular about their disks. They get a little weird sometimes. You don't want to get 2 or 3 months down the road, everything loaded and just the way you like it only to get a little pop up saying your copy of Windows is pooched and wont let you access anything.
Cheers
Rob
The answer is always "PEBKAC!