I told my clinic about my job, it just means that they ask me slightly different questions, and occasionally I chat with the nurses about political developments and stuff.
http://www.praedstreetproject.org.uk - That's the only clinic I know that's specifically for sex workers.
Yes, in a good clinic you tend to be treated better (for one thing you're more interesting than the average 'worried well' or the 'I seem to have caught x yet again' clients!)
If you do rimming or scat, you'll also want the Hep A vaccination, and some places can get appallingly sex(uality)ist about that one if they don't know you're working.
The male equivalent of Praed Street is the Working Men Project, but this has changed its services somewhat over the past year or so. Another service from the founder of WMP is in development.
The West London Centre for Sexual Health (i.e. Charing Cross Hospital in Hammersmith) has code word based clinics for female, male and trans escorts and porn 'talent' (e.g. if you're a male or trans escort, you ask the Health Advisers about the Diamond Clinic and you get fast tracked).
There's also a hospital north of the Thames in West London which has a similar system, but I don't remember which or the magic words.
I know another couple of hospitals in East London and North East London also have specialist services, and one in South East London certainly used to but a change in the services the PCT funded about two years ago may have altered that.
Outside London, some places have specialist services, most don't. Some services that do exist can be rather more based on 'helping' you 'exit' than on harm reduction.