private a levels are the same as normal a-levels and you will be accepted into university with them but the only difference is that you will be studying independently, as for the exams you may have to pay for them yourself if you are not enrolled at a school or college and they can be very costly.
there are a number of places where you ca take exam centres where they allow private candidates to take their exams and sometimes you can be able to take them at a school as a private candidate.
for me I did Edexcel A-level Math and I was able to find an exam centre nearby where I live.
there are also online courses that teach you A-Level subjects such as UpLearn however I'm not sure how efficient they are as I haven't had any experience with them but you could have a look online to see reviews of what past students have said about it, you could alternatively hire a tutor to help you with your learning but this may cost a lot.
Also if you are planning to apply to university after your gap year, then you may have to apply to UCAS as an independent student unless you are going to apply as part of a school.
hope this helps