"National curriculum" doesn't apply in the same way to Key Stage 5 (over 16s). There are subjects offered at A level that are also National Curriculum subjects (Maths, English, Science, History etc) but there are lots of others that aren't (Economics, Philosophy, Sociology, Law, etc).
In my subject, at least, when they went over to linear A levels, the three main exam boards agreed on what the main content would be, so while there are some small differences between their approaches, the students learn very similar material, take the same amount of exams, etc. I don't know if this is the case for all subjects.
Something else to know is that the old modular system of A levels is being phased out as the subjects gradually transfer to two year linear courses with the exams all at the end. The rationale was to give more teaching time and to prevent students endlessly retaking modules to boost their grades. Students can still take A/S levels after a year but many schools are dropping them due to the expense of running two courses.
In order to teach without QTS the school (if state maintained) has only to show that they cannot recruit a qualified candidate (I don't think anyone ever checks on this); academy schools (the majority) can recruit whom they like and so can private schools.
In practice the majority of teachers do have QTS as without it, it could be hard to move jobs and you might get paid less. But it is not really a requirement, although a older colleague told me the government has been saying it will be soon since the 1970s!