It's torque that matters rather than outright power. Some seemingly powerful engines are short of low-down grunt, and this can be frustrating. I haven't driven a 2.0 petrol Ford - try to find one in your price range from a dealer and get a test drive.

I'm not sure what age of car your budget buys you: but if you go for a pre-DPF diesel like the Kia mentioned above, or a VW or Skoda with the 1.9 TDI engine, you could find something reliable. Skoda Octavias used to be available with the 1.8 petrol engine, but they may be hard to find.

What you really need is an 'old school' 2 ltr petrol engined car. Sadly not too many of them around on forecourts thesedays - its mostly little turbo-charged ones, medium-sized Japanese offerings that you have to thrash to get decent performance or the latest de-tuned 2 ltr offering from Mazda (which might've been fine [it appears to be the equivalent performance of a 1.8 petrol] if it were also available on older models in your price range).

Your £4.5k budget unfortunately limits you to older cars if you intend on going for one with a bit of poke. You could try the previous gen 1.8 petrol Toyota Avensis - very reliable, reasonable performace but good mpg (see HJ reviews), but isn't going to handle anywhere near as good as a Ford or Mazda. As Avant says, it'll be the test drive that is the deciding factor - all the rest can be gleaned from online research, reading reviews/comments (as here) and speaking to friends/family/work colleagues.

Ideally the beemer 2 ltr would be the ideal choice on performance, but whatever you could afford (not much) would be a lower-spec model for 'newer' models or old, less reliable ones if you wanted more toys or more space (say a 3-series Touring or 5-series) with that engine, which has its faults.

Never easy when the budget's tight. You might find a worn-in car might be better than a (tight-engined) brand new one on mpg (my Mazda3 1.6p was 10-15% less on the mpg in its first year than after), so the Mazda6 might still be worth a look. I would also nail down what the minimum spec needs to be acceptable to you (particularly with the toys), as they can make a large difference on the price, even in older cars.

When you go for the test drive, try and do so on a reasonably warm/sunny day (difficult I know, especially as its now Autumn) and turn the A/C on to a low temperature so you can see what effect it has on engine performance when you're giving it a bit of welly. If you or any family members have a bad back or don't like a firm or bouncy ride, take them with you (this will also gauge the performance with people and boot loads [it may be worth taking some stuff to see the effect]) to see if they prefer the ride of one car over another - Mazdas mostly have firm rides (though not as much as German premium cars on huge rims), as do Hondas, though I think Toyotas may be a bit better on that score.

Best of luck.