2nd gen Note auto uses an apparently fragile supercharged 1.2 along with an also apparently fragile CVT, so best avoided

Agree that the CVT is dreadful (not just reliability), many reports of issues and no one will fix them. But not seen reports of issues with the 1.2 DIG-S. We had on in a manual Note and it was very economical and the car was incredibly spacious and versatile. But I will add its not a patch on the similar VAG 1.0 TSi like we now have in the Fabia. The TSi is just as economical but has loads more torque low down (max torque on the DIG-S is at 4000 rpm, the TSI has more at 1500 rpm and it just keeps coming) plus the TSi install is way more refined.

We have often said that the Note with the Fabia powertrain would be unbeatable for us.

The Mazda 2 automatic has used a torque converter auto since the 2nd gen came out in around 2007. AFAIK the first gen cars used the same single clutch automated manual (Durashift) as fitted to the 1.4 Ford Fiesta (6th gen) and Fusion (the 1.6 versions used a t/c)

With a £6k budget you'll get a late 2nd gen facelift car (around 2013) and yes, they are very reliable.

As to whether one would make a good alternative to a Jazz?, that really depends on what why you were looking for a Jazz. If you wanted one because of the huge interior and boot space (for its size), along with the flexibility offered by the magic seats then no, a Mazda 2 wouldn't be a good alternative!. It has a much smaller boot, less interior space all round, and does not have the flexibility. If that is what you wanted, a better alternative would be the 1st gen (2005-2012) Nissan Note, which is of a very similar size to the Jazz. Doesn't have the Jazz's magic seats, but the rear bench does slide back and fore and the cabin is very spacious for the size of car. The auto uses a n/a 1.6 and t/c auto and they are very reliable (2nd gen Note auto uses an apparently fragile supercharged 1.2 along with an also apparently fragile CVT, so best avoided).

The Mazda2 is certainly the best to drive (handling wise), though though the gen-2 cars in auto form are not quick by any means, just adequate.

The main 'issues' of the cars generally are space and ride quality - the nice styling doesn't make for a large amount of interior or boot space, especially compared to the Jazz, which is far more practical, as is the Note - both being more 'boxy' in shape.

A former colleague owned a manual Mazda2 gen-2 and it was shod on relatively low profile tyres despite being average spec. As an occasional passeneger, I found it to be firmer than I experienced in my Mazda3 (at the time shod on 'average' 55 profile tyres).

As such, best to make sure if the OP looks at one, that it come shod on at least 55 profile (preferably nearer to 65 profile) tyres or that they are happy with the ride quality via a long test drive over a range of road surfaces.

I might've added the Hyundai ix20 or sister car, the KIA Venga 1.6 petrol TC auto, as that is bigger inside and the boot, but its engine and especially its TC auto box is rather dated in design and, as the Real MPG figures show, gives around 31mpg, as opposed to 37 for the 1.3s in the Mazda2 and Note, and 44 for the Jazz1.4 CVT.

The smaller, less high-sided Hyundai i20 / KIA Rio in 1.4 auto form is 10-20% better than their bigger sibblings on the real-world mpg front, so may be worth a look.