A Qashqai wouldn't be the worst option and you could definitely get one for six grand. They're smaller than you'd think and very easy to drive. If it's too big you could look at a Nissan Juke, it's the next size down.
If you go to the Autotrader website, you can search by price, make and model, and filter by ULEZ compliance too.
I tend to prefer Japanese or Korean cars, generally. More reliable. So: Toyota, Honda, Subaru, Kia, Hyundai. If you want the crossover style you could look at Honda CRV, Kia Soul or Sportage, Subaru XV, Mitsubishi ASX, Toyota UrbanCruiser.
I'd actually steer (!) you away from Audi, Mercedes, Mini, BMW or VW. They're no more reliable than anything else, and statistically less so than most Japanese makes. And when German cars go wrong, fixing them can be expensive. (I know this from personal experience, having had two Audis in the past.)