NFC has nothing to do with the carrier or the SIM card, it's an internal function of the phone. Read
this Wikipedia article - it's accurate. I have a 10 year old SIM in one of my phones that I had my carrier activate for a while when the big phone was away for repair (long story short - an OTA update killed the phone).
If the SIM works, and it's the right size, or it fits with an adapter, it works. You could ask your carrier if they could switch your account to the new SIM, but if it were me, I wouldn't bother. (The only reason I got a new SIM with the new phone is that the old SIM was too large.) I'd put the new SIM away, so that if the old one went bad I could call them and ask them to switch it then. (We probably won't be using SIMs by then, though - they tend to last a very long time. I still have a working one from one of the first GSM carriers. Can't use it, because the carrier no longer exists, but the SIM can still be written to and read from.)