When you qualify for a vaccine and go online to book it, you find there are vaccination centres and pharmacies all over the place giving the vaccine, it very much depends where you live and how lucky you are that there is one at a reasonable travelling distance.
If you wait for the GP to get in touch then you will get one closer to home, possibly at your GP surgery although usually a group of practices have merged to make one place available between them for vaccinations as was the case with me.
RHC - if you weren't anxious about waiting then your GP would have sent you a letter to make an appointment much closer to home. I suggested Solid go online because of his and his wife's circumstances and the fact he said the GP isn't very reliable.
@Solid_Air 2 I'm pleased to have helped.