My first port of call here, would be get the car Hunter aligned. Prices for Hunter wheel alignment can vary wildly, from £25 just to do the front tracking/toe angles, to around £75 for the whole lot to be done, though from what I can gather, there are some places that will charge over £100 to do a full Hunter alignment, which seems kind of crazy. Believe it or not, some of the best places to get proper wheel alignment done, are crash repair/bodyshop places who do wheel alignments day in day out as part of returning damaged vehicles back on the road.
I had a full Hunter wheel alignment done in March last year and my back end was well out, especially the rear nearside. The front was out too as about 18 months before, I'd replaced the front struts/coils and droplinks. The whole alignment cost me £60. The ahh, Technician, had a helluva job trying to get it within specs and almost managed it perfect, but it was still about 1 degree negative on the rear nearside, everything else was spot on.
I haven't had any more issues, though I have just last Thursday, had the front wheels re-tracked on a Hunter because I replaced the tie rod arms. I'm currently doing a 500 mile round trip, did 235 miles yesterday and it's as straight as a die.
Depending on the cars mileage, I would agree with above. Possibly worn banana arms or rear control arms or toe links.
I have directional tyres on my car, but I regularly swap the backs to the front and vice versa, obviously ensuring I just move the same sides back and forth to keep the correct directional pattern.
As for the part worn tyre thing, personally, I'd be fuming. The part worn tyre industry is a minefield. As a matter of interest, have the tyres the dealer put on been stamped correctly?
https://www.tyresafe.org/tyre-safety/part-worn-tyres/part-worn-tyres-law/
Personally, I'd never put part worn tyres on my own car, you have absolutely no idea where they've come from or whether or not the vehicle they came off was involved in an accident. I realise this wasn't your choice. Personally, I think it's a bloody cheek putting them on without consulting you first.