Just to add to what matt isserstedt posted, for the sake of completeness:
There's at least one other method of achieving the same (highly desirable) result as the telescopic feedscrew in his option 2.
My P&W toolroom lathe has a two-storey or double-decker cross-slide, where the lower slide (which includes back-to-back male dovetails top and bottom) is either clamped (for normal work) or driven by the taper attachment (for taper turning).
The feedscrew support bearing, handwheel and all, are part & parcel with the lower slide rather than, as would be usual, with the carriage apron, so the whole shooting box moves in and out when taper turning. To get power crossfeed to the feedscrew the pinion on the feedscrew is really really wide, looking more like a straight involute spline than a pinion gear, so the saddle gear can drive it regardless of position.
I haven't got the lathe running at present but it does seem to me that if power cross feed and longitudinal feed are both available at the same time (something I've never tried, but I know some lathes permit it, and looking at the clutches they don't appear to be interlocked), the lathe would turn a steep taper, maybe 45 or even 60 degrees depending on the ratio of cross feed to long feed. It would do this over a much longer travel than the (frankly rather tiny by modern standards) compound slide. I doubt however that you can control which quadrant it is - I think it is probably correct for taper bores but wrong for male tapers.
With the double-decker cross slide arrangement described above (maybe even with a telescopic feedscrew - I'm not familiar with them although I've used DSG lathes which have that arrangement) you could theoretically then engage the taper attachment to adjust the steep taper either side of the 'default' angle.
Way to go, 2 axis power feeds plus taper attachment all working away at no extra charge on a lathe designed before WW2 ...
Could be handy, has anyone tried this?