The NHS should cover everything in terms of doctor's appointments, emergency room treatment, other hospital treatment, prescriptions, birth control etc. and should normally be your first port of call in all cases. There's no such thing as having 'pre-existing conditions' on the NHS because it covers all the treatment you need for every medical condition you've ever had, for the entire time you are in the UK, all for free (well, you pay for it out of your taxes).
Birth control is free to all (including visitors who don't have free access to NHS treatment) and you can either get it from your NHS doctor, or you can go to a Family Planning Clinic and get it there. It doesn't really matter, as it's free either way.
Generally, private coverage is only really useful for getting things done faster or for accessing rarer drugs/treatment that are too expensive to be covered on the NHS, or for elective surgeries that are not offered through the NHS. For example, if there's a waiting time of a few weeks or months for an operation or to see a specialist, you can skip the waiting time by 'going private'. Often you will see the same surgeon or doctor or specialist using the private coverage as you would have on the NHS, but it will be done more quickly.
For example, both my grandmother and my godmother needed hip replacements. My grandmother had it done for free on the NHS and got the operation within 3 months. My godmother paid for private treatment and had the operation within a couple of weeks. I think they both had the same surgeon.
Usually though, if you want to use your private insurance, you need to go through an NHS GP first and then get a referral through them for private treatment.
If you bring your US medical records with you, you should be able to register with a GP and get a referral, although you may find there are long-ish waiting times for appointments on the NHS, which is when private insurance can come in handy if you are able to use it for them.