Clearly home made - yet by someone with access to a foundry and reasonable array of machine tools - this ingeniously constructed lathe is of the plain-turning (i.e. non-screwcutting) type and lacks a slow speed backgear. Surprisingly, the centre height is only 21/4" over the bars and the distance between 8" - that is, if it was possible do that, there being no taper (or even a hole through the headstock spindle) and only a short, possibly truncated No.0 Morse, in the tailstock. As part compensation, the frames that form the headstock and tailstock are in cast iron, milled flat, and then drilled as a set to take the bed bars - these almost certainly being off-the-shelf ready-ground stock.
All fasteners and stock materials are imperial - yet the ball bearings in which the headstock spindle turns are metric (probably because they just happened to be "handy"). The 3-step headstock is machined for round belts, with a V-belt used from motor to countershaft.
Clearly a much later addition - and not of the same quality as the rest of the lathe - the  motor mount has clearly been cobbled together from a few bits of wood and a gate hinge - though it is suspected, from some existing remnants of parts, that the original drive might have been by a foot-operated treadle. The countershaft drive pulley - and the leadscrew handle - look as though they might have originally been used on something like an industrial sewing machine. One interesting point concerns the cross slide, this being formed by the use of a slotted block running in two keyed plates - simple and easy to make, but virtually impossible to set up to give a smooth, free travel yet little play….
There are, unfortunately, no identifying marks on any parts of the lathe.