Hi Dan,
for 2K you should be able to find a pretty decent Colchester, Harrisson, or other light-industrial lathe, if you're really lucky you might get a Dean Smith & Grace, a CVA (British clone-ish of the Monarch 10EE) or a Holbrook, the DSG is more production, the CVA and Holbrook more toolroom - all will come with niceties like quick-change threading gearboxes, power feeds, and above all build quality to last longer than the (Chinese) warranty!
There are useful threads of choosing a lathe, basically they come down to "what do you want to do with it?" and "how much time can you put into it?"
I bought a scruffy but sound Holbrook C13 (now 60 years old, at least another 60 in it) which needed a VFD adding as it's 415v 3-phase only and did a basic clean fix and refurb, it's a joy to use compared to the Chinese imports, even those costing 2 or 3 times your budget, and cost me less than a well-tooled secondhand Chinese mini-lathe...
The fact it weighs 2 tons and needs 415v may have put the other bidders off, but a hiored 4x4, sturdy trailer and scaffold poles / 4x2s to roll it on made moving pretty simple, a hacked VFD (cheap older unit off Ebay) delivers 415v from 240v house mains and adds braking, more control of the speed etc. which now ranges from 4 rpm to 2500 with 3 HP behind it!
DO remember though, the tooling can cost more than the machine, so try to get as much as you can with the machine - it can make an older industrial machine much cheaper even in the short term if it comes with (e.g.) 3 and 4-jaw chucks, a collet set, centres, quick-change toolpost and holders (I'll never go back!) rather than a bare brand new machine with just a 3-jaw chuck.
