Excuse the suggestive title, I don't intend to offer you any NSFW material.
Now that we have that topic out of the way, I must inform you of my problem.

I just bought a new, Brass nut for my P bass copy, and after removing the old one (and breaking it in the process) I put the new item in.
After this, I have realized that the height of the nut is too low (despite measuring three times, and checking against the original) and the string touches the first fret.

What could I use to prop the nut up? and would using different things end up changing the sound?

I've been told to use cut up bits of old credit card, is this a wise idea?
Thin wood shims are better than credit cards in my opinion. Glue the nut to a strip of wood about the same thickness of a credit card (could be a little thicker) and then trim the wood to the size of the nut. Put the nut in the slot (don't glue yet) and restring. Once you have the guitar set-up with the new nut dry fitted, check to make sure there isn't any buzz on the open strings and that strings aren't too high at the nut/first fret area. If everything checks out, you can slack the strings, move them off to the sides and out of the way, and glue the nut in place.

If there is buzz, you didn't use a thick enough shim. You can add another shim the same way and should be fine. Chances are with your first shim thats credit card thickness or thicker, you will gain enough heigh to clear the buzz/first fret contact.

If the string action feels too high around the first few frets, your shim is too thick. Slack the strings, move them off to the sides of the neck, and take the nut out. Sand it on a flat surface, being careful to keep even pressure distribution so that the nut sands level. Sand a little, check the height, and repeat until you have it proper.

It sounds like you have the original nut height, so you should be able to measure to determine just how thick of a shim you'll need and dodge the guesswork in doing so. You might want to leave it a little on the high just to be safe (you can always sand later).

You're not going to notice much of a tonal impact.