At the Jenner Practice, where I have been a patient for many years, you can generally get an appointment within 48 hours (I believe that this is a NHS requirement of any group practice of GPs).
You can queue up at the crack of dawn (not pleasant if you're really ill!), you can phone up, and now you can book on line (you have to register for this).
The Jenner "releases" appointments on a daily basis and if you know the time when this release happens then you can book on an automated phone system. it seems to be at several times during the day and night. Once you have got your head round the system then you should be able to get an appointment quickly.
Suggest you try and talk to the receptionist at a time when they're not too busy, or transfer to the Jenner!
Maybe a member of the admin staff at either of the Practices could explain the appointments system.