My advice would be to search repeatedly, not just on Autotrader, but also on eBay, Car & Classic, etc. Maybe stay away from Gumtree and Facebook Marketplace, as those are full of scammers.
I bought my 2012 Jazz EX with 98k miles on the clock for £3k, last May. Clean MOT history, effectively rust-free, and this was from a dealer, so I think you should be able to get a reasonable example for £2k from a private seller, especially far away from London.
Prices can drop a little in the run up to Christmas, and if you go for a pre-facelift model (2010/2011) and a lower spec (ES/SE, rather than Si or EX), I think you are likely to find some good examples. I wouldn't go for the 1.2l model, if you can avoid it, as that tends to be quite slow. Manuals tend to be cheaper, but if you are set on an automatic, go for the CVT, rather than the i-shift transmission, which seems to be regarded universally as awful.
The anti roll bar drop links are cheap and easy to replace. Rust is the big killer, and what you really should look out for. Mentions of corrosion on MOT advisories are a bad sign, as that means that any rust is likely progressing beyond the superficial. Obvious places to look out for it are the wheel arches, the sills, the suspension mounting points under the bonnet, and the rear subframe. Some surface rust (discolouration, but with solid metal underneath) is fairly normal with a high-mileage vehicle, and you can keep this at bay with undersealant. If the metal crumbles to dust, that is generally bad news. Have a look inside the boot, too, under the liner. Rust or moisture there means that the boot seal or the light seals might have failed.
Look also for signs that the previous owner might not have looked after the car (e.g. repeatedly failing MOTs on worn brakes and tyres, broken shock absorbers, etc. - the odd advisory here or there is fine, but if there is a pattern, maybe that owner did not treat that car very well). If you know someone who knows their way around a car and what to look out for, take them with you when you inspect the car.
And yes, be super wary of any car that has been recently undersealed by a dealer. This can be a big red flag. I spotted this on a 2014 model, and that immediately told me to move on (the inside of the car also had a thriving mould ecosystem, which was another bad sign). Do a quick history check on any vehicle, and buy a proper check (with carvertical or a similar website - you can find plenty of discount codes online) before you make any purchase, to make sure there isn't pending finance or any hidden history.
Just a few quick thoughts, I'm sure you will get plenty more from other members.
Keep in mind that Jazzes are quite plentiful. Don't be pressured into a deal if you aren't convinced!
Good luck, and please keep us posted!
