Latest update. I've got the car back now, with my new used transmission (although the dealer kept referring to it as a gearbox, and when I asked if he meant transmission, he said "yeah whatever you want to call it").
Anyway, he also fixed the seat belt after I mentioned the specific laws around selling second hand cards and the road traffic act.
However, he refused to give me any documents as a receipt of work that was completed, nor would he even tell me who fixed the car. Apparently he doesn't even have the paperwork yet, the car shop is yet to send it through to him, but said that he would send them through when he receives them.. He literally walked away from me as I was talking to him..
As was mentioned earlier in the thread - keep written records of everything - I've done just that, sending him a follow up email "capturing" our conversation that he would send through the details of what was fixed, and I also enquired about any further warranty on the replacement transmission/gearbox (I'm pretty confused at this point what was actually fixed). I feel like this "warranty on the repairs" is a bit of a shot in the dark, I couldn't find anything in the Second-hand Vehicle Dealers Act 1995 that talked about warranty repairs - and as I type this, he's replied asking what I mean by warranty of the repairs.