Just a word of warning...........some surgeries might be ignorant of the rules and have some sort of crazy idea your wife isn't entitled to register and has to be a private patient.
First of all - to confirm, the no recourse to public funds thing means benefits. It absolutely does not include the NHS, being able to send your kids to schools etc - she is completely entitled to all these as your wife.
We however had the little battleaxe on reception coming out with all sorts of rubbish, telling us we couldn't register, sending us away to get more documentation (We took proof of address, passport, entry visa etc already). I checked two other surgeries and they both said "no problem, just come register", but we wanted to register at the one I went to.
In the end, the chief receptionist said she would allow us "just this once" to go around "the rules". I should have complained about her really - but can't be bothered, there are more important things in life to worry about.
That said, it's just a warning some might think they know it all, but frankly don't, and doctors receptionists do tend often to be a certain breed of power deprived types who you have less chance of getting past than the bouncers outside Stringfellows if you turned up in trainers, Y-fronts and a string vest