If I were in such discomfort I would just go into boots and buy some, or get it online, rather than waiting until next month to see a GP.
Vaginal estrogen has been available to buy for some time now, and I know you shouldn't have to pay for it but on the other hand you could have requested it on the NHS years ago rather than waiting until things got this bad.
It may even be worth putting in a prescription request for vaginal estrogen to your GP practice in the interim, many offer electronic communication and they may be willing to issue the prescription without waiting for a face to face appointment.
Additionally, is there a reason for only wanting to treat your vagina rather than your whole body?
Whilst not everyone chooses to use systemic hormone therapy, this should always be an informed decision and not just the default due to lack of knowledge or access.