I doubt the doctor will cut it out immediately, but I know when my hubby had a recurring problem of this nature, he made an appointment with the GP for a day or two later and had it removed under a small local. What the doctor may do is put a dressing on it until that can be done. When it has been removed, it will probably be sent away for routine testing - just to make sure it's benign.
If your GP does do it there and then, I should tell you that when hubby went to have his done, he was most definitely not looking forward to it - the removal or the test - he is a such a total wuss about these thing. However, he said it was easier than having a filling done and he didn't feel a thing beyond the anaesthetic injection. The doctor did the cutting with a sort of hot wire which cauterised the wound at the same time, so he didn't even need a stitch. He said there was a bit of a funny smell, and he felt a bit of pushing and pulling around the mole, but that was it. The test came back negative, which the doctor said is mostly the case.