MODEL ENGINEER (M.E.) threads fit the bill.
5/32" 40 t.p.i. 5/32" is 0.1562"
Available from Tracy Tools in the UK. You don't have to buy sets, they will happily supply new or secondhand items.
There are other model engineering shops in the UK that sell taps and dies in the ME series,
As the name suggests used by model engineers, clock makers and engineering instrument makers for donkeys years.
I have used a few myself in past years.
Of interest to photographic types like us, there have also been in use a long time ago, Royal Microscope Society of London (RMS) also known as Society Thread, and Royal Photographic Society Threads for Lens Flanges.
Here is a copy of Model Engineer Threads I was given years ago.
As you can see, there are two series, 40tpi and 32 tpi The tapping drill sizes are in Imperial Fraction, Number Gauge and Letter Gauge, just in case non engineering types are not familiar with the drill sizing shewn.
I hope this is of some use to you.
Not trying to hijack this topic but, I am curious to know if the Model Engineer Thread Sizes are peculiar to the UK.
Are they a British invention? I would love to know their history.