When I had AZ jab I wasn't told anything about not driving for 15 minutes, and everybody just had jab, walked out, got in the car and drove straight off, so I presume nobody was told that. Other friends have had AZ jabs in other centres and nobody has had to wait or been told not to drive.
Advice on the leaflet says "COVID-19 Vaccine AstraZeneca has no know side effect on the ability to drive and use machines. However, side effects listed in section 4 may impact your ability to drive and use machines. If you feel unwell, do not drive or use machines"
Friend had Pfizer jab at same centre and had to sit and wait required 15 minutes in the centre.