It depends. It's a key question to ask when you're visiting.
Most of the ones I know follow the same literacy guidelines and many use the same reading schemes (e.g. phonics) however I remember visiting a school when DS1 was 4 and being less than impressed by a school that seemed to have a hotch potch of really old reading books, including a whole load of 'Letterland' ones which, at the time, seemed to be being phased out. (Having said that, they could well have been following the same NC?)
Think some senior schools teach International Baccalaureate rather than NC?
IME - they use the NC as kind of a 'guideline'. Probably quite a good target for flogging books, as they tend to own multiple reading schemes and 'pick and choose' between them as it suits individual children.
And there is the independent schools inspectorate, whose website escapes me right now.
They don't have to follow the NC. What their end goal is is to get their kids into the right senior schools for them. This means they need to prepare them for various 11+ and Common Entrance and Scholarship exams.
There is nothing wrong with the NC, just how some schools choose to apply it, especially with respect to KS2 testing. Most teachers are trained to follow the NC. Therefore, there will be a lot of NC in independent schools, but they can pick and choose what to do, and what to do on top.
Independent schools are inspected by the Independent Schools Inspectorate. If they receive state money, ie in foundation stage, they will also be subject to Ofsted inspections. They can also be inspected by the local authority if they have boarders.