Many thanks for all of your help. After much deliberation, we decided to get a higher mileage (front-wheel drive!) Audi A1 from a relative. It's been serviced regularly (perhaps too regularly at every 3,500 miles!) so we're not worried about the mileage, especially given the fact that it was a lot cheaper than the 1 Series.
Here are some replies to some interesting comments....
@TheFairyCaravan Not all young lads drive like twats and neither do all young drivers have lots of knocks and scrapes.
Absolutely! Given how strict DH is with driving 'perfectly,' I'm pretty confident DS won't be driving like that. It's a shame that the majority of 17 y/os, who drive poorly and play bumper cars on the road, create a stereotype for the minority.
@PulpPixie My first car was £7000. I wasn’t about to drive around in a rust bucket. It was a Ford KA.
The rust bucket point is very valid. We didn't want DS driving around in something embarrassing. More importantly, given the fact that the budget was £6,500, I'd have preferred to have gotten a more premium car, rather than a 'standard' first car for exactly the same price. Absolutely nothing to do with checking privilege if not spending the same amount of money, surely?
@NoIDontWatchLoveIsland *Totally baffled by the number of people buying 17 year olds their own cars costing thousands.
what happened to teens working to save and buy their own bangers. Im mid 30s, when I was 17 the girls & guys who wanted cars fought for the best summer jobs after GCSE, and worked evenings and weekends in shops or cafes to earn the money. Parents didn't hand out free cars! Spoiled much?*
If I can afford to make my child's life slightly (a lot!) easier, why shouldn't I? His parents won't be around forever - and I don't want him to face needless difficulty whilst we are. Nothing baffling about understanding that A-Levels are a difficult time and that he shouldn't have to spend his time working a job so he can get to college without having to wake up at 6:20 for a 1.5 hour bus journey.
Simply because our parents (I can't speak for you but mine certainly did) made us work to buy our first car, it doesn't mean that is the right thing to do. You ought to step out of 20 years ago because logic has taken over. If parents can afford to purchase their child's first car - it's their choice whether they wish to or not. If not, that's life: the child can work for their car.
a BMW? Are you bloody kidding me? That's a luxury branded car. There are adults working in essential occupations earning barely enough to feed their families & you think it would be nice for your teenager to be pootling round flaunting his parents wealth in a BMW. He will wind up cocky and entitled, if he isn't already. If you hand your youngster everything for free what incentive is there for them to work & get themselves a decent job to afford these things?
Oh boy. My DS is far from cocky and entitled and has been brought up to accept whatsoever he is given, thank you. If he was so cocky and entitled, he'd have suggested the BMW, not I. If I hand my son everything for free for now the incentive for him to work hard is being able to continue being able to afford things he likes by getting the career he wants. He's extremely aware that DH and I won't be around forever to bail him out, and it's with this fact in mind that he does work hard for the "decent job" you mentioned.
A lot of people on this thread have spoken about their first "banger" of a car. I don't want my DS to have this memory. I want my DS' to remember his first car as something special. I want it to be something he can look back on as something his parents got him because we felt he deserved it. If you wish to provide your kids with the memories of how hard they had to work for a 'banger' - go for it. But going for this route turns the transition from childhood to adulthood, marked by getting a car, into a harsh reality check rather than a smooth transition. Again, parents' choice.
I also think that forcing DC to get 'bangers' encourages carelessness when driving - particularly parking. It pushes them to think that it's acceptable to bash their cars a bit because they were cheap, rather than trying to actively improve their parking skills. Seems illogical to me. 
@Alsohuman Why would you buy your son a BMW when he wants a (far more suitable) Honda Jazz?
I don't think he actually wants a Honda Jazz. It was more loud thinking than anything else. In reality, he'd accept anything we were to give him gratefully since he's not really the demanding type - has made the last 16 birthdays quite hard I can assure you! 
BMW idea was more for reasons outlined above than for 'showing off' value (as others would suggest).
Many thanks to all for opinions. To those who have DC nearing 17, best of luck to them; wishing them many happy miles ahead of them!