Mine's sorted now
I didn't sign the agreement over the weekend, got no reply to my message on the app, so went to drop off my car this morning feeling a bit worried not knowing what was going to happen... and the showroom was deserted at first, but eventually a member of staff came out (who remembered me from a test drive a few months ago) so he started looking up my records etc to read the message I'd sent, but as he was doing that I got a phone call from someone in the service department there anyway (as he'd realised no one had replied to my message), and when I said I was actually there he came straight out to chat to me in person.
He explained that the date/time gets put in automatically when they generate the agreement, so I bought the excess insurance policy online from my phone at that point (insurance4carhire, £12.79 for 4 days cover, I'd already managed to speak to them to check that it would cover this situation) and then he regenerated the agreement so that it showed the current date/time, and I accepted it. He did have to come back and help me another 2 times though, when the key locker wouldn't work (needed rebooting) and then the PIN to drive he'd given me didn't work either (seemingly because of the regeneration of the agreement), but all sorted in the end lol.
From what he said, they are very easy going about the condition of the car etc so he said I didn't need to take photos etc (even though the Tesla app actually said to at that point) but could if I wanted to. And the car actually was in good, clean condition, inside and out, and had the type 2 cable (and even the Tesla granny charger bag) in the boot.
So I'm happy with that, feel like I'm covered well enough now, and it ended up being a pretty good experience at the SC (Crawley/Gatwick) in the end.