If money is no object then there are a lot of advantages to using a DC motor for a lathe. The ability to have continuously variable speed control is a huge advantage for a lathe. Compared to other machine tools a lathe has the specific problem that the circumference (and thus proper cutting speed inevitably changes as you work. Consider turning a 30mm bar down to 10mm, the circumference changes by a factor of 3 so any fixed cutting speed is going to be a compromise, the alternative being changing gears mid job.
It is also very useful to be able to fine tune cutting speeds on the fly when cutting threads or turning difficult materials. Being able to precisely adjust cutting speeds also lets you get more effective capacity out of a given motor power.
The alternative is an AC induction motor, these are cheaper and don't require electronic control but by the same token don't allow for speed control in single phase.
There isn't much to choose between the two in terms of startup torque. Larger AC motors are generally split phase which gives high starting torque.
You limitation on power will be down to what your supply can deliver. Single phase motors generally go up to about 4hp but this requires a 240V 20A supply.
To give you some sense of power requirements a 500W lathe with a DC brushless motor will cope fairly well with drilling holes up to about 16mm diameter and just about manage 25mm holes with care in metals with good machinability (ie leaded or sulphurated mild steels or brass) and cuts up to about 1mm at a single pass.
For a small lathe (say 120mm chuck) a 1200W motor would probably be ideal and that's about the limit for a domestic power supply.
In general DC motors tend to be a bit more compact for this sort of power requirement than AC induction motors also the ability to add speed control greatly simplifies the transmission design, although having a few gears is still an advantage.
My experience is that for a small lathe running off a single phase supply a DC brushless motor is a lot more usable and efficient in terms of overall production rates than an equivalent AC one. A 3 phase machine gives the best of both worlds but that is not an option here.