The duplicated thread now appears to be closed, so to repeat my answer here;
If you use your favourite search engine on "Stanley Plane Thread Sizes", there is a wealth of information, all concurring that it's a current non-standard size, some say 3/16"x28, others No.10×28, neither of which help you, because it's no longer a standard, if indeed it ever was one.
Easiest solution to me is to buy an M6 thumbscrew off eBay or similar, and turn off the thread, almost all the way to the top. The remaining bit will allow a nut to be Loctited on and turned down to form a collar.
The rest, when the thread has been turned off, is almost spot on size to cut a 28tpi thread.
(or, as mentioned previously, the thumbscrew on mine is a rattling good fit, so you might just get away with an undersized 1BA @ 28.3tpi)
When you've got those sussed, try Singer sewing Machine threads next, where you will find several 3/16"x28 were used..
https://www.singersewinginfo.co.uk/screw_threads
From the Singer tables and catalogues, it becomes obvious that this thread was used on several machines.
My own walking foot patcher is a 29K4, so quite old.
It's the only one for which I have parts manuals.
The two suitable candidates are screws 274 +locknut 1539, both of which seem to be unobtainable, or a 515 wingnut, also no longer listed anywhere.
See plates 1046 & 7742 of this pdf
https://www.universalsewing.com/images2/parts_lists/all/74nfn30o.pdf
There is another one which is used to lock the rotating head; on my machine, it shares the same thread size.
On later machines it comes up as a Singer part No. 109535 – SS535W, but I don't know if they changed the thread.
These are available from College Sewing, amongst others.
https://www.college-sewing.co.uk/ss535w-head-rev-bush-stop-thumb-singer.html
N.B. I don't know if it's the same thread, as no=one ever seems to publish the specs. but I suspect it is.
Bill