I wouldn’t backdate it. I’d speak to your boss and verbally give your notice and let them know that you have a letter to formally confirm that you are leaving. Then I’d ask if I could leave a day early, if they say no, then I’d speak to the other firm and let them know that you will be working through the night and therefore unable to work on the Friday.
TBH it would ring alarm bells with me if they insisted on you coming in on the Friday, because firstly it would mean they have a lax approach to hours and secondly, if they have a driver leaving them, they’d want that driver to finish planned work. When we take someone ask, we ask them when they can start and that’s the day they start. I certainly wouldn’t expect someone who had been driving all night to come in and do work, which is what an induction day is.
As for the letter, simple, just say that you are giving a weeks notice, you have enjoyed working there ( in case you wnat your job back) and that it’s time to do something different/work for more money/get ADR/hiab experience).