In my experience, if you're going public, you see your GP for your initial visits (checking you actually are pregnant, then the first bloods etc.) and they'll refer you to the hospital. Depending on the hospital, they'll either book you in for your first visit early-ish, or you'll keep seeing your GP until around 18w. If you end up with the latter and you want to do DS testing, you'll need to go ask your GP for that around 10-12w. Then you go back, get the results, and see midwives at the hospital for most visits. You'll usually only see an OB on certain visits or if you're high risk.

Be prepared for them not to tell you a THING. My very first BB post was pretty much exactly that - "my doctor isn't telling me what I'm supposed to do, when is this supposed to happen? When do I do this? Why isn't the hospital contacting me yet?" It's like they expect you to know what to do without any guidance. Hopefully you end up with someone better and my experience was just bad luck...