Yep. I would turn up at my normal place of work at the alloted time, and then travel in their time.
I would also then travel outside of rush hour to reduce my travel time
Yep. I would turn up at my normal place of work at the alloted time, and then travel in their time.

I know personally I wouldn't (and have flat-out refused to in the past) be working in the evenings out of principle - they've asked you to make the journey, therefore that is work time, you can only do so much work in what remains in the day. When I was doing work for the NHS early on I was keen to get the work done and would put as much time into it as possible, but that was without the commitments and family you have, and that grew quite tiresome after a short while to the point that I wanted to stop it, and did after some firm discussions with my managers and Trust board.
temporary, HMRC would probably define this as less than 3 weeks, then it becomes your fixed office.
Which in turn means you could argue you have been relocated if over 3 weeks duration.
What does your contract of employment say.
My company states first hour of travel to a site other than the office is classed as commute. Anything over that is working time.