Drill Chuck for AT350WL lathe

I am looking for a drill chuck for my AT350WL and would appreciate some recommendations. I think I need one with M2 but the one from Axminster Tools is not available. Thanks in advance.

Sounds to me as though you need to join your local Woodturning Club - where you will get plenty of advice and guidance.
Certainly you need a chuck with an MT2 arbor.
Axminster sell plenty of Jacobs chucks, keyed and keyless, but without the arbor - which comes separately.
Talk to your local store and they will be able to sort out suitable parts. (Customer Services will do the same if you phone them.)
You can also find plenty on the internet.
Happy turning, Philip.

How do I find a local woodturning club? I did try googling but can’t find one in East Sussex near enough to my home!

I Googled Woodturning Club near East Sussex - Google Search and came up with many hits. Not knowing exactly where you are I can’t make a recommendation but suggest you try the link. It is unlikely that you will have a Club on your doorstep but I think you will find one within 20 miles that suits your requirements.
Hope you are successful.
Happy turning. Philip.