Hi there, all,
Put in some more workshop time today.
I'd pondered on the best sequence of operations and realised that, in the later stages of machining, the work-piece was going to be held by a narrow spigot one end while the other spigot, the thread and the tapered section were machined. To give the job some support, I fitted a piece of brass in the tailstock drill chuck and drilled it from the headstock, drill #24, a running clearance for the diameter of the two spigots. Here's a photo, sorry the camera auto-focus decided it preferred the background!

I had originally planned to hold the raw material in a ⅜" collet to minimise overhang but, for reasons cited in my earlier post, that didn't allow the tools to reach the job! So I used my 'Sunday Best' three-jaw instead. Further thought revealed that I needed to win some clearance between the top-slide handle and the tail-stock body (remember, the tail-stock is set over in preparation for machining the tapered section; I didn't want to disturb that setting.) So, I extended the tools further out of the QCTP holders than I usually have them.
I started by machining the long spigot, first with a right-hand knife:

Then cleaning the corner with a 'Mini-Thin' tipped grooving tool:

Next, I machined the bar to the diameter of the major diameter of the tapered section and, using the 'Mini-Thin' tipped grooving tool, marked the extent of the threaded portion:

I then reduced the diameter of that section to ¼" and threaded it ¼" x 40 tpi ME using a tail-stock die-holder. I had to refine the undercut at the end of the thread by repeated applications of the grooving tool. Before proceeding, I checked that the shape so-far achieved fitted nicely in the sub-stage bracket of the microscope. Sorry, forgot to take photos of that operation. I then trimmed the long spigot back to the correct length.
It seemed advisable to remove as much metal as possible with the job still supported on the end of the raw material bar. So I put in two further grooves to denote the boundaries of the tapered section and the other threaded section. I next withdrew enough of the bar from the chuck and supported the long spigot using my tail-stock steady while I plunged the sections of the bar whose diameters needed to be reduced. I reduced what was to become the second threaded section to ¼".

Again, I had to apply the grooving tool to refine the depths of the grooves as this work progressed. When the short spigot section of the job was finished to the required diameter, I removed the job from the chuck, sawed the work-piece off the raw bar and reversed it in the chuck, holding on the long spigot. I was relieved to find that it revolved true.
Proceeding carefully, I threaded the appropriate section ¼" x 40 tpi ME using a tail-stock die-holder. As before, I had to refine the undercut at the end of the thread by repeated applications of the grooving tool. (The QC tool-post really shows its value when repeatedly swapping tools like this.) I also used the grooving tool to clean up the shoulder where the short spigot meets the threaded section - if you examine the photo carefully, you'll see that I overdid this a bit! I then carefully trimmed the short spigot to the correct length.
Because of the ravages of anno domini, I was a bit tired by this time and I should have quit while I was winning! 'Winning'? Instead, I decided to go for broke and machine the tapered section. There was just sufficient room at the start of the cut for the right-hand knife tool with the end of the short spigot engaged in the tail-stock steady.
Now, I do know, from folk-lore, from all the books I've read, from all the forums I've browsed and even from some practical experience, that you do have to be very careful turning tapers. The sharper the taper, the more care is required and this taper is only about 8° included angle.
BUT, I was tired and when it was almost done, I decided to take 'just one more cut'!!!!

So, this is what I achieved through my day's labours:

Well, that last cut took off about twice the meat that it should have done so the taper protrudes a few thou through the polariser unit rather than being flush. I might be able to shim it, time will tell. Also, some of the threads that seemed fine while work was in progress, now seem to have tightened up, maybe some fine swarf in the threads. So, I plan to sleep on it and do some fitting and fettling and see how things turn out.
Maybe this one will be declared a trial run and I'll machine another - it should be simpler second time around!