Do you think what you're missing is London itself, or the idea of London?
Trains and Turkish supermarkets are things you find in any city in the UK, and a lot of towns, so it's interesting to me you mention those but then you say the real problem is moving to your home city.
I can totally relate to that! I would never move 'home'. I know people who would, but for me, purely emotionally, it wouldn't work. I think there is tendency for people to say that if you move, you should move to somewhere you already have roots. Actually, it can be the worst thing.
I have never lived in London and would never live there, so take my advice with a massive 'she's a country bumpkin' pinch of salt. But, most of my extended family and both of my brothers are in London, and I read your posts thinking how familiar what you were saying seemed, having listened to them. DP and I also moved away from where we used to live because, for us, it was too expensive and hothouse-y (we were in Cambridge, which isn't London prices but is something like 2nd/3rd most expensive after London, and the education system is likewise quite nasty). We thought about moving to my home town, or her home town, or other places we knew. We decided instead to take the plunge and move to an area we'd both loved from going on holidays, but didn't really know. We had some very small connections to the area.
We've been here just over a year and it is home. It's amazing. I still walk out of my house and feel so grateful and glad I live somewhere I love so much. Because it wasn't a familiar place, we had to work to get to know people, but that effort paid off, and it's been absolutely amazing to have the experience of being welcomed into an entirely new community.
I would say that, if you possibly can, consider moving away from your 'home city'. It sounds as if it is holding you back and stopping you from building a new life.