Looking on the web the Taig / Peatol spindle nose is 3/4-16tpi UNF so you could make the adapter on the lathe if you have 1/2in BSF taps and a 3/4 UNF die.
The spindle nose starts as 1" AF hex which is 7/32" thick but no real reason why that shouldn't be 1/4" really. Bit more for a spanner if a chuck gets stuck. Or made from round. Then there is a plain diameter register 0.750" dia and 3/32" long The spindle length from the face of the hex is 1/2" long and threaded up to the register. The o/d of the thread is 0.748"
It's usually possible to keep taps axial and square with a centre in the tailstock and dies with the nose of the tailstock. So you would start by drilling and taping for the flexispeed nose. Just tap far enough for that as this might turn out to be handy later. The screw it onto the spindle and turn the od to 0.750 up to the shoulder.
Get a split die if you can and check the size of the die holder. Just past WW II these stopped being made slightly over sized. If they are very close to the right size which they usually are they benefit from being opened out by 0.010in so that the die can open up a touch when the screw that goes into the split is firmly tightened. It's a mistake to open them up too much as the die might break. The screw on the holder should have a point on it to match the die and the handles can usually be unscrewed so that the central part can be held in the lathe. With a bit of care this allows a bit of adjustment and as many dies now cut undersized the thread may well turn out to be correct when the die is wide open.
When dies are used like this it's better to use the nose of the tailstock to square them up rather than a tailstock holder as opened up they aren't central any more. Just keep the tailstock hard on it while cutting and backed off when it's reversed to break the chip. Sane with the tap and centre.
You might be able to machine the adapter out to take Peatol / Taig collets. The taper for them starts at 1/2" dia but I'd guess that the flexispeed thread is too long to allow that to be done on an adapter for the chuck. A longer adapter could be made though. The taper is about 5/16 ~~ 3/8 deep. Best way to do that is get one and use blue to set the taper. A DTI on the compound slide edge helps see how much they are being adjusted.
Unimat ? I did have one. Parts are rather over priced in my view and quality no better but it's another option. Rather more expensive if 3 and 4 jaw, face plates and collets are added. Blank collets are even available for a Taig which can be sized to suit and split with a hacksaw.
Some might not like the aluminium jaws on the 3 jaw but I found them to be fine and easy to machine. They can be trued up enough to make collets a bit pointless really. Have a suitably sized thin penny washer handy and machine a shallow recess at the very back of the internal face of the jaws. Grip the washer in it and press it home and square from the back of the spindle and firmly tighten the chuck. Then lightly skim bore the jaws. Remove the washer and skim off the slight step that will be left just past the recess. Main problem is that the lathe needs to be turning dead parallel. If the inside grip of the usual 3 jaw chucks is turned that's not so important as they can only be used to grip short things or longer things with a centre or something in the end. Same when they are reversed.
I tapped the washer and screwed a bit of bar in so that it could be pulled up against the recess from the front and then unscrewed. If I remember correctly the washer was about 1" od.
John
PS I think I saw a broken flexispeed on ebay with a 4 jaw fitted – casting broken by over tightening one of the bearing clamp screws after a lot of wear. The painful aspect of that design where they are just split on one side.
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