Congratulations on the move!
Your wife should be able to sign up with a GP, sick or not.
I signed up with my local GP within a month of moving to England. They had me make an initial appointment with a nurse so they could get my information into the system. The appointment wasn't as intensive as a physical - she took my weight, height and BP, and asked me a couple questions (if I smoke, drink, etc). That was it.
That said, every practice perhaps operates a little bit differently. Maybe your wife could try calling again, or finding another GP in the area and trying them.
Good luck!
ETA: I made this post under the assumption that your wife has no NHS history. It may well be different if she does, as she would (I'm guessing) already be in the system.