Having just had a blood test (I’m registered with the Jenner practice, BTW), it seems that during these COVID times, there are no walk-in clinics anywhere in the Lewisham health district area.
You need to visit the centralised web booking site (or use the phone number) indicated on your blood test request form, and pre-book a slot at one of the centres currently offering blood testing facilities.
Website is Book a blood test | Lewisham and Greenwich - but obviously you can only book a test if you have the request form from your GP.