The obvious has already been said - you need to get an NHS or private dentist to refer you to the hospital in the first place.
Beyond that, I can only tell you that I had some dental work done in the hospital a few years ago, under a local. My treatment was called an apisectomy whereby the very root of my tooth had to be drilled out and filled via direct access through the gum, and required a couple of stitches afterwards. It was absolutely painless, beyond the sting of the first couple of shots of anaesthetic into my gum. They put several shots in, in various places and all I could feel was a bit of pushing around in my mouth.
For an extraction, though, I'm not so sure. I've only ever had one done under local, and tooth broke as it was pulled and the whole process was very painful (even though the dentist kept saying it wasn't hurting me - how would he know?), so I've only ever had extractions under sedation since then (well, just one, but that was enough). I say sedation - it was the white stuff injected into my hand that knocked me out for the few minutes it took to remove the tooth, and was in addition to local anaesthetic. That the same thing?