At current (and increasing) rates, first doses will be finished way before September. More like April/May - assume say 50m left to do at 4m per week, will be done in 3 months or so. Once we have virtually everyone vaccinated they won’t need to insist on proof of vaccine because the numbers that haven’t had it won’t be material anyway.
It is incredible how the NHS, army and Volentary services have achieved this to date but I fear the supplies of vaccines will prove to be its downfall.
With regards to vaccine passports it's almost certain to be adopted the problem is the uk cannot do this in isolation all nations need to have them in the way passports are standard worldwide the cost and logistics are huge far better to have a reader chip in with the vaccine!