My advice would be to wait until you have your lathe and see how it goes without a chuck. There are several reasons for this.
Firstly, you might not need a chuck anyway.
Secondly, if you buy a chuck now then it limits your choice of lathe to the thread on the chuck. Adapters are available and M33 is getting to be a standard now but better to get the lathe you want and get a chuck to match if you decide you need one.
Thirdly, there are plenty of good, solid lathes available especially second (etc) hand, but there are some rubbish ones too (woodzone and clones - SIP, Clarke, Lumberjack , Vevor, Einhell and loads more, all the same cr@p, just badged differently). Not all Clarke etc are cr@p, but I am thinking of one model in particular. So rather than spend money on a chuck that you might not need, put it towards a better lathe. Then you are less likely to want to buy a better lathe if you 'take to it' and want to upgrade - beware, turning can be very addictive and very expensive if you catc the bug.
Finally, after you have been turning for a while, that is the time to look at chucks, jaws etc, as you will have a clearer idea of your needs.
Regarding the lathe, as I said there are loads of options. Some are not so popular now (Tyme, Arundel, Turnstyler, Myford and others) and whilst they are good quality, the spindle threads may be difficult to match these days. Axminster do a Hobby range which are well thought of by owners, although Axminster used to describe their budget kit as suitable for about 100 hours of annual use with few, if any, periods of continuous use. That's 2 hours per week, which gives you an idea of build quality. Their Trade range are much dearer.
Second hands brands you could consider within a reasonable price range (under say £600) are Record, Union graduate, Denford Viceroy, Poolewood, probably other turners will recommend more. I have a Myford ML8 - old and limited on capacity and features, but well built and will go on and on. It is already about 70 years old. I also have a graduate - more capacity, heavier build, rock solid in use, and was the choice of the professional until recently. and I have a Viceroy, again larger capacity than the ML8, and solidly built. I have avoided recommending beasts such as Wadkin - real professional kit but again not easy to get chucks etc for these days. I recently hd the option of a Wadkin RS10, absolute beast, 7ft 6 ins between centres, total length 11 ft. I could have bought it for £500 but - what on earth would I do with it?
K