Hi Luke
The bed looks really sound a few close ups would be nice
¼ H.P. should be powerful enough for a small lathe, my Myford has a ½ H.P. which is well powerful enough even though mine is a little bigger see
A History Of The ML7
1725 R.P.M. is a four pole motor at U.S. frequency of 60 Hz (ours are 1475 ish at 50 Hz) you should have enough torque at slow speed (Brit bikes again lots of bottom end grunt )
The main problem with single phase motors is starting 'em up, my ½ H.P. puts a fair blip on the mains for ¼ second or so when it's starting.
1" to 2 ¼ what size is the driven (headstock) pulley? with an identical opposite (driven) pulley with four step settings you would get ratios from 1 to 2..25 (max speed about 3900 R.P.M) to 2.25 to 1 (min speed about 780 R.P.M.) I think this might be a little fast
With regard to your headstock runout it sounds like your headstock bearings might need to be tightened a little bit.
I'm not sure about the arrangement on an Atlas 109, but on mine it was simply a question of re-shimming it (1 thou less ) and then re- bedding them, lots and lots of oil with some graphite in it put the 3 jaw on and turn it by hand back and forth until it stops binding.
I'm probably teaching my granny to suck eggs here if you've owned a few B.S.A.s and Triumphs you know the routine already
3 jaw chucks are not that accurate if you can get less than about 3 thou' runout from a piece held in the three jaw you are doing well, it usually doesn't matter 'cos you are turning from a common centre (Eng again) so everything turns concentric to itself.
I have nothing against Hondas (My first bikes were both Hondas) or Japanese bikes in general, they are relentlessly reliable and get you to work almost always in all weathers.But,,,,,,
Brit bikes, yes they are like Harleys in a way ( probably more power from a smaller motor). just something a bit special about them "cantankerous" as you put it .
A Rolls Royce. A Vincent Black Shadow. An Electraglide,I have ridden in or on all three and loved each in a different way.
Pick your fav or perhaps you have one of your own P.M. me if want to.
so post some pics of the lathe buildup (you can post 'em directly BTW just inc 'em in the post)
Regards to you and your Good Lady
(hope she doesn't hate us TOO much)
Mark