Question Author
Thanks for your answers, I'm going to write to HMRC.
My previous employer 'didn't see <me> remaining in the UK', hence no need to give me a P45 as far as they were concerned.
I had the same problem with Adecco several years ago. I temped several months in a financial company which was having difficulties. They eventually had to make several tens of thousands of permanent staff redundant. Of course, we temps (several hundreds of us at the site where I was working) were let go of before HR started going after the permanent staff.
Adecco didn't give us new jobs or P45 (so we couldn't claim JSA either). I managed to find another assignment rapidly, but because it took me 3 months to get a P45 from Adecco, I had a much lower net salary than normal... and I had to move house and fork out a new deposit to be closer to my new job (I worked shifts and the commute in the early morning was horrendous) so it was really bad timing!