I feel, after reading this, so lucky to have a surgery with a triage system which guarantees a same day appointment with either a nurse or a doctor. The nurses will refer you to the duty doctor or nurse practioner/prescriber if necessary. Open Saturday and Sunday mornings too. 13 doctors, and lots of nurses and nice receptionists.
Nailit. If it is bleeding you do really need to have it sorted. my husband had a similar problem. It wasn't a skin tag, but a benign purple lump that bled. The doctor removed it. I really feel you should see another doctor. Best wishes.