In late March I found that VeriFly only accepted one of us. The LHR T5 agent was not surprised when I handed over the paperwork! For vaccination, they want to see the NHS app on your phone - because you could have forged the print out. Of course, you could have forged the image displayed on the screen of your phone too! My partner could not get data access whilst standing at the check-in. So I had to make a HotSpot on my phone for her to connect. We then had to repeat that the next morning in JNB.
Whenever I travel, I always print flight confirmation, car hire, hotel bookings the lot. At the end of the trip it goes in the recycle bin.