When you arrive in the UK you will need to find your local doctor / GP and register with them as soon as possible. You may find they have overlapping catchment areas so you can choose between 2 or more different GP surgeries, especially in London. You could try asking neighbours what the GPs are like, or how many appointments are available. It's quite common in the UK to wait 4 + weeks for a non-emergency appointment. If the surgery has multiple GPs you lose the continuity but there may be more flexibility.
Once you've chosen one, (and make sure it's NHS, there are private GPs in the UK, especially London, but unless you are seriously rich, they're not an option for every day care, and private health insurance doesn't usually cover them) then go into the surgery, ask the receptionist if you can register. Take along proof of address, ID like passport, and in this day and age, anything you have that proves you have a right to free treatment here. NHS number would be useful if you've had one previously.
As soon as you are registered ask for a 'booking in' appointment with the midwife, and explain you are 30 weeks and have just moved to area. Hopefully you should get an appointment quite quickly - preferably within the week. Take your partner with you as they may be asked lots of questions on family medical history.
Maternity care is usually midwife-led, assuming there are no more serious issues, with you seeing the community midwife at your GP surgery, increasingly frequently as you get close to due date. You'll probably be sent for a scan and they may want to redo tests like bloods, glucose tolerance etc.
If you have any serious medical problems, or the pregnancy is complicated (multiples, previous issues, diabetes, heart issues, anything picked up in the scans) then you will see both a community midwife and be referred to either the nearest hospital, or possibly a specialist hospital if required, to be "under" an obstetrician and specialists from any relevant medical areas.
If you have an uncomplicated pregnancy then your birth options are:
(1) at home, very popular in the UK, you can hire pools, the midwife brings gas and air which is the UK pain relief of choice in labour. You may be able to get pethidine too at home, not sure, but if you'll want an epidural then you'll need hospital. Usually you'll get 2 MWs during the labour, and if things aren't going well you'll be transferred to hospital - local or specialist depending. You won't necessarily have met the MWs before, it's difficult to get continuity except in rural areas or if you pay for a private MW but NHS birth.
(2) midwife-led birthing centre - a halfway house between home and consultant led birth. Again for uncomplicated pregnancies only. You go when in established labour, and a midwife will deliver your baby. They're usually attached to hospitals in case a transfer is needed, but they tend to be a bit more relaxed and about non-intervention. Birthing pools, gas and air, not sure about more advanced pain relief.
Or there is Hospital care. Can be either for complicated pregnancies, or because you prefer to be in a hospital. Advantages are consultants and doctors ready to assist with forceps, c-section, epidural etc and paediatricians ready to help a sick baby, disadvantages are that it can be more clinical and a less natural feeling birth experience, and may lead to more medical interventions. There will be birthing pools and gas and air still and with careful planning you can still have a relaxed birth in a hospital. Your MW will be able to advise on which hospital you will be going to.
Hospitals try to keep the number of c-sections down, so generally you will only get one if medically required - either in an emergency or because of complications identified earlier in the pregnancy.
Worth knowing that in the UK pregnancy is calculated from the date of last period, which is different from some countries. You are classed as 'overdue' from 40 weeks since last period, and at 41 weeks they usually intervene. This is different from other countries.
Culturally, and especially in London, there's been quite a movement towards natural births. It's not standard, as it is in say France, to give most women epidurals, though you can request them. You usually start with paracetamol, then gas and air at hospital and a birthing pool/bath, then possibly pethidine later. French woman I know find this barbaric!
After the birth if you and baby well you are both well then you might be discharged the same day, or next morning. It's unusual for people to stay in more than 3 days unless there are serious complications. Then a MW visits you at home the first few days, and then switches to a health visitor (nurse specialising in babies and families in the community). You'll have a "6 week check" of your health at GP or possibly in hospital, when baby is 6 weeks old.
Join a local NCT course (book well in advance) to meet people in your area who are also having first babies, you get trips to the labour wards etc too. Hypnobirthing is very popular and works well.
You can go private but the costs for a full private birth are extortionate - It think it's in the tens of thousands. Private Health Insurance doesn't cover maternity care.