I do not know how your healthcare coverage is in the uk so I cannot answer the cost issue.
Depending on the size of the mole and how it needs to be removed determines whether he/she will remove it right then and there or not.
I know that here in the u.S, insurance will only cover the cost if there is a medical reason for it to be removed. Like, if they think it is cancerous or poses a risk to your health or if it causes pain etc.

yes, it depends if the mole pertrudes from the area, or is it almost as flat as the surface of the area. If it sticks out, than the doctor may have it removed at the clinic. He would probably apply a numbing solution and cut it off with a cutting instrument, but if the mole is deeper in the skin, you may probably be sent to a surgeon and probably be given a few treatments where he'll slowly remove the pigment with a laser. The second option is quite costly. Doesn't matter where you live, unless you have some kind of health care insurance, you will be charged. I don't think it will cover this though since it is not a medical reason.