Hi Penny I've never had a nail removed but I'm an NHS Podiatrist so I've done this procedure in the past.
As I'm NHS ., I have no clue about prices but I'll give you a run-down of what happens.
(Might be different in different places though)
History - any allergies, medical conditions. Have you had a local analgesic before (similar to the dental injections but without adrenaline)
The injection (which is the bit people worry about most) goes into the base of the toe at either side.
I had to have a 'practice' every Friday for 6 weeks at college, then a written and practical exam in 2nd Year. They aren't the worst injections IMO.
Our lecturer used to make us lie flat ("You can't faint lying down" so ask to lie flat , breathe normally)
They should inject quite slowly, you'll feel 'tingling' which is the analgesic starting to work. Once it's numb, all you'll feel is pushing and cold, not pain.
Tourniquet (little rubber ring) goes on to keep the blood out.
Drapes and antiseptic to pre-op.
Nail is loosened with a thin chisel type instrument then cut and removed (sides or whole nail, whichever it needs)
Phenol, (strong cautic) applied, timed and neutralised. Stops the nail matrix regrowing the nail.
Tourniqet comes off, it'll bleed- a bit/a lot. Depends.
Dressing (bulky) . Wear open shoes, you'll have a numb toe.
I'd have a painkiller (non aspirin) before it wears off. (About 2 hours numb time)
Keep your foot up that night.
They'll see you back in a couple of days and advice you on dressings. (Our patients self dress and salt water bathe the toe)
Post -op pain varies but most people say it's ok (phenol is a good painkiller)
Takes about 6-8 weeks average to completely heal, some people take less.
Success rate 90% +.
Worse patients - young men 
Good Luck