No problem at all! I did exactly the same thing. In fact I had to change my name in the new passport -- since I had my maiden name in the old one --before I could apply for the Police clearance at the passport office. And this was just before P4. So, P4 had my married name on it.
Just make sure you write or call the NVC with the information about your new passport and send them/email them a copy of your new passport and your police clearance.
Best of luck!
