Hi, Jessie!
You are entitled to NHS care as soon as you become a resident, one who resides in the UK. Once you arrive, you will be a resident. As soon as you can, go to your nearest health center and register. It's a good idea to take something with you that proves your address. Tell the receptionist that you will need prescriptions soon. You will need to be seen by a nurse for intake before you see the doctor. Once you see the doc, your BC and repeated prescriptions will be easy to obtain.
I just registered today at my local health center. I see the nurse on Thursday and the doc on next Tuesday. The whole interaction took maybe 5 minutes.
I don't know how much your scrips will cost. Much less than COBRA payments.
Chris