Since 2017, these are the Commissioning guidelines for certain procedures in common with many practices across the land.
Procedures
Treatment of Minor Skin Lesions
including benign pigmented
moles, comedones, corns/callous.
milia, molluscum contagiosum,
seborrhoeic keratosis, skin tags
including anal tags, spider naevus,
warts, xanthelasma and
neurofibromata, epidermoid/Pilar
(sebaceous) cysts,
Lipomas
Thresholds
Only commissioned where there is:
Suspicion of malignancy
OR
Obstruction of orifice or vision
OR
Previous Infection
Will be routinely commissioned under the
following circumstances:
Severely functionally disabling and/or
subject to repeated trauma due to size
and/or position
Source
https://www.stokeccg.nhs.uk/generic-publications/page-documents/299-excluded-restricted-procedures-policy/file
I would still try for another GP appointment stressing the current bleeding as a articular worry, if only to get advice on how to proceed privately.