You are buying on a budget and think a BMW is a sensible option!

Large and reliable (or cheap to repair) family car on a budget = Mondeo. Very reliable, spacious and comfortable. Decent TDCi engine, cheap parts and lots to choose from.

With £4k to spend, a diesel is too much of a gamble: what you might save in fuel can so easily be wiped out by an expensive engine failure).

I'd go for a Mondeo 2.0 petrol (lots to choose from) or a Toyota Avensis, Honda Accord or Mazda 6 (all very reliable with petrol engines).

What annual mileage will the car do? Diesels are really only worth it (even on second-hand cars) if you do at least 20,000 miles, maybe even 25,000+ to cover the extra purchase/service cost and potentially (especially with second-hand models where you don't know exactly its driving history [e.g. if for part of its life it was used on short journeys]) much higher costs associated with component failures.

As has been said, for older used cars, forget the badge and go for reliability and at least reasonable running costs (note that 'premium' [i.e. German or little-known foreign makes who have high-cost imported parts] also cost much higher to insure as well as service/maintain than other 'mainstream' makes) as well as the driving abilities of the car you choose. Also, for the same reasons, go for cars with less electronic gizmos - they have the potential to be expensive (out-of-warranty) failures, so if you don't need them, look at a lower-spec model (it'll obviously cost less to buy as well).

Best of luck.