Is it because your GP isn't available or because they're booked up? If the latter, maybe go in and say you'll wait as an emergency appointment until they have a few minutes for you?
If you send it recorded or registered delivery by the deadline it will be counted. I'll chuck you the money for that if you need it.
I had a similar problem in that I have recently changed GP so she's only seen me twice. Due to the recent IT hack she didn't even have access to my records last time I saw her. And when I requested my letters from my rheumatologist, because for no known reason they hadn't been sent to me, the most recent one was in error and about someone else who had a bad back, plantar fascitis and two small children. FFS. It had my name and NHS number but was not about me. Those are two of the diagnoses I actually lack!
So I might well have to appeal (eventually) due to these fuck ups. Doubt that would come through before my blue badge etc are due for renewal in September.