I would not worry about mileage over condition and service history.
From the forum there does not seem to be any significant issues with the 2009 Jazz, so it's more a case of use - did they run it up the curb every time they parked, scuff every car in Sainsburys, or was the dog often sick in the back.
There may only be a small pool on here, but you should get a good car.
I suggest telling some small traders exactly what you want, and they may be happy to assist. If they know they can turn over a car immediately for a few hundred profit, then joy all round.
BCA is probably best for ex-lease/motability 3 year old cars.
Evans Halshaw has quite a fast moving range if there is one nearby, although choice is much wider at £5k.
The best private deal won't last long, so be prepared to buy it when you see it.
You can probably try for a couple of £100 deductions on obvious things like tyre wear, scuffs, and time since last service; if you try walking away you can always return 5 minutes later if that doesn't work and 'it's the one'.
Sold a 4yo Mini Cooper the morning it went live on Autotrader to the first caller, a couple who travelled 150 miles after agreeing to buy it on the phone. Told them the phone was ringing off the hook, didn't really know if they were going to turn up, but the whole family arrived at 8pm. They did not bring proof of insurance so could not test drive it themselves.
So take ID, proof of 3rd party cover, and online banking or cash - and be prepared to act quickly.
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TG