Your baby is seen by a paediatrician in hospital before you are discharged.
Then you'll be visited by the community midwife at home for the first week-10 days (depending on how things are going).
Then she'll sign you over to a health visitor. Where we are (also Wandsworth, but the Putney end) they have two HV clinics a week (one morning, one afternoon) at the GP's surgery that you can make appointments for, plus one afternoon drop-in session at a local health centre where you can just turn up. You can also leave them a phone message and they'll call you back at other times. Typically for a first baby you might see them every week or two for weighing etc. at first, but it's up to you and if there's no cause for concern you needn't go that often.
Your baby will need to be registered with a GP. They will give you his NHS number at the hospital when he's born, and you'll get a form to give to the GP when you register the birth. Effectively the GP will take him on as a temporary patient right away and add him to the practice list when they get the form.
If you aren't happy with your existing GP then it's as well to look into changing now. Is it the whole practice you don't like, or just one particular GP? You could ring round other practices in your area (from phone book) and see if they cover your address. Each practice will have a fairly limited geographical area from which they'll accept new patients, but in London it normally works out that several of these areas overlap so that you have a choice.
The baby's first check (after the one in hospital) will be at or around 8 weeks. At our practice the health visitor does part of it and the GP does the rest, but I don't know if that's standard. Again, the practice might make the appointment for you automatically or you may need to call and make it yourself -- you need to check.
Then, apart from vaccinations if you are vaccinating (generally with a nurse rather than a doctor) and seeing the health visitor at whatever intervals seem appropriate, the next official developmental review is at 8 months and generally with a health visitor. They do make it clear what to do if you've got any developmental concerns before that, though.
You should have a post-partum check at or around 6 weeks (our practice does them at 8 weeks so they can see you and the baby all in one, but I think that's unusual) -- again with the GP.
If your baby is ill your first point of call if it doesn't seem urgent is the GP. They will refer you to a paediatrician if it seems appropriate. Or, if it's urgent, you can take them to Accident & Emergency Many hospitals have specialist paediatric Accident & Emergency departments where your child will be seen by specialists. There is also the NHS direct number (will be in the booklet your health visitor gives you) that is very helpful if you don't know what to do. You ring and explain the problem and a nurse rings you back, talks through checklists and advises you on what to do next (get GP appointment / go to A&E / wait for X hours and watch for any of these symptoms, etc.).