If it's just you, so don't need space for significant luggage or DC, I'd go for a VW Up, Skoda Citigo or Seat Mio (they're all pretty much the same car). All small petrol engines but read up on congestion zone/ULEZ rules to check. FIat 500s have a poor reputation. Look cute but not good cars.
Very low insurance and if you go pre April 2017 (check exactly when) the road tax will be no more than £30 per year.
Look on Moneysavingexpert for how to get the best insurance and breakdown price. There's a few companies that provide the equivalent full service breakdown cover of the big names for about £50/60 per year, so about a third of the cost of the AA etc.
You can test out insurance costs etc by finding the reg plate of a car you like the look at and putting your details into comparethemarket.com or similar, even if you have to estimate. It will save all your details so once you've set an account up, it will be quick to do. You'll also get a feel for which companies are cheap for you - they all have a particular demographic they want business from, and price accordingly. Be aware that your insurance will be more expensive after you've passed your test, rather than being a supervised learner, if you get a car before you pass.
If your budget is up to about £8k, I'd maybe go up to about £6k and keep a chunk back to pay your insurance up front and for any repairs. What people tend to forget with cars is that they need servicing, tyres and brakes etc doing every few years and a small chance of a larger repair. Put a few hundred by for these sorts of costs each year, and if you don't spend it, it's a bonus.