Hi Antonia
I felt like you a few weeks ago so completely understand your question.
I rang my surgery to first of all ask if they were still undertaking routine blood tests (I have one every three months) and yes, they were, but I made the appointment with trepidation.
On arrival, I could enter a small lobby but the surgery door was closed. The receptionist came on a speaker as a disembodied voice asking me my business. I said who I was and why I was there. I was then asked my date of birth (presumably as a check) and told to enter - the door slid open.
Inside the waiting room there was.... no one. No patients at all. This was not that the doctors were on holiday, but that 'hot' and 'cold' hubs had been set up elsewhere, so patients did not attend this building.
I waited a minute or two, chatting to the Receptionist (from a huge distance!) and my name came up on the screen directing me to a nearby room - so no stairs or other bits of furniture to touch.
I opened the door with my scarf and inside the phlebotomist was fully-gowned up, including a visor, mask, apron, gloves and could barely be seen!
She took my sample and I hopped out again a few minutes later, again using my scarf to open the door.
Exiting, the door was opened for me by the Receptionist, so I touched nothing.
To be absolutely sure, I had in the car (which was opened using a different part of the scarf) some hand sanitiser which I immediately used.
Once home, I put my (clean, grrr) clothes into the washing machine.
I feel completely at ease returning for another test if the same precautions are being used.
Reading other comments here, it seems that most people who have been for a blood test are satisfied that the correct precautions are being undertaken.
Go and have your test and try not to worry!