So. The flatbed they sent wouldn't take the car because the wheels wouldn't fully disengage and the driver wasn't comfortable using skates on it. The car was completely dead and while it was in neutral and could be pushed out of traffic, the shift buttons wouldn't work so the car couldn't be put in park. After a while all of the flashers died and went out (I couldn't turn them off with the hazard button). When the first truck arrived we discovered the wheels were now resisting turning past a few degrees and would then mush to a quick stop. We couldn't push it any more. It did not feel like a firm lock like you see in park.
So after he called his people and I called Jaguar assistance again, they sent a more conventional two truck with an arm to lift the rear wheels (under the tires) and Dollys for the front wheels. However, the driver wanted me to sign a damage waiver... so I called Jaguar and they called the dispatcher for the tow company. After about 15 minute of holds while Jaguar talked to the tow operators, Jaguar eventually told me I was going to have to sign the waiver if I wanted assistance. I did have them put in the record that I objected to this and expected Jaguar to cover anything that went wrong with this (I know this probably isn't legally valid but I needed to get the car recovered and had been there for over two and a half hours in the mid-day Texas heat).
So the driver carefully loads up the car and we move on to the dealer. He did a great job BTW! I had tried to call the dealership but they were closed for the 4th and had no emergency contact option available. I figured their gates would be closed, and of course, we get there and the gates are closed and positioned so close to the street there is no place to leave the car. They have no contingency plan in place for receiving after-hour breakdowns apparently.
Sooo. I call the Jaguar assistance line yet again and they agree I can get the driver to take the car to my home to stay there overnight and then I can call for another tow tomorrow when the dealership is open. (The doors won't lock and the handles won't retract so I don't want to leave it in a public place other than the dealer).
Altogether it ended up being just shy of 4.5 hours from the first call (about 15 miles from home) and getting home.
I'll send another message once the dealer has the car and can tell me something.