FWIW I'm not biased, because I have two Axminster lathes but recommend the Record one. My second choice would be for the Axminster one and in fact I looked at one last week and it's difficult to fault it IMHO. I use my chuck all the time and the range and versatility of accessories e.g. C jaw etc. is very important - this investment continues even when you change the lathe. So matching what you might do and comparing the chuck as a system and not just one expense is something to consider. I would stick with Axminster for this because I think their range of options is more extensive, YMMV. The same chuck can be used on any of these lathes.
Hope that helps...
You can of course use any chuck with the right thread for your spindle (and you can get adaptors for most). So you don't really need to think of buying the Axminster because you like the Axminster chuck. And in fact that particular Axminster has the slightly less common M33 x 3.5 spindle.
I looked at both the Axminster AC355L and the RP DML320, but then actually bought the Axminster AC370WL. A bit bigger than the 355, and with a swivelling head, but cheaper because it has mechanical variable speed rather than electronic (but with 10 steps, that seems good enough to me at the moment). And though it came with legs, I am using it bench mounted. And it has the 1"x8tpi spindle that is more common, so easier to find chucks to fit.