In my area (Luton and Dunstable), when you need a blood or urine test from GP or consultant, they can give you a form in advance which lists which tests required (the forms are identical, regardless who gives you one). You can then take this along to anywhere which takes blood tests, and there are a number of drop-in centres. I go to a different surgery from my own which has a drop-in centre for bloods and you typically wait about 10mins. The other nice thing is you can go in with multiple forms from different doctors, and they just need one jab to do all the tests (and sometimes just a single phial). The only bizzare thing is the results are only visible to the doctor who requested them, and not other tests done from the same phial at the same time required by another doctor.
See if you can get a blood test form, and if you have the option of local drop-in centres.