On the drivers medical web sites,not many you can book on line,the one i found,it comes up with the location,and time and day you want,then a message comes up saying not possible,i looked at doctors on wheels and drivers medicals.com.
Any recomendations for the Somerset/Wiltsire area,or further away is ok,if needs be.
I did mine through Doctors on Wheels at Avonmouth Toby.
It used to be at the truck stop but then they moved it to the other end of the estate.
I just done mine with my local health centre,
just picked up the dog&bone and made an appointment.
Handed over the dosh that my firm reimbersed, job done.
It really aint that hard. 
perhaps it was just that they were fully booked on the day you wanted 
btw, here’s a plug for the cheap medicals thread
viewtopic.php?f=13&t=19955