Gaby, I haven't had a baby in the States, but am originally American and was used to the U.S. medical system until I moved here five years ago. I wanted to add my two cents to try to reassure you that it's not all doom and gloom as I have had a very good experience with the NHS with two pregnancies: one relaxed, low-risk home water birth, and my current high-risk pregnancy with identical twins.
One thing you will have to get used to is that the approach to pregnancy and childbirth is far less medicalised here than in the States, which I welcomed. I know I would have been viewed as a crazy hippie earth mother had I attempted my home water birth in the States, and now with my twins I am grateful to be in a country where the rate of c-section for twins is around 50% as opposed to 75% in the States.
If everything is going well and you are at low-risk, you will not have very many appointments and will see midwives instead of doctors. You will also have more options for the birth. Depending on your hospital, there should be rooms equipped for water birth or active birth, or even a midwife-led unit, if you feel comfortable with any of those options. Epidurals and other drugs are not pushed to the same extent they seem to be in the States, and many people attempt to get on with gas and air before trying anything else.
If there are complications or you are at high-risk, you will be seen by a consultant, although it won't be quite the same as private care in the States. Unless there is a particular issue that the consultant needs to be involved in directly, you might find yourself seeing other doctors on their team. So whilst you would be monitored closely, you wouldn't necessarily have one-to-one care. I try to view it as comforting as I know if there was a serious issue I would see the consultant himself straight away.
In any case, good luck with the move and the childbirth! And yes, start by registering with a GP straight away, who will get you in the NHS system and point you in the right direction to get a booking appointment with a midwife and get things rolling at a hospital.