Myths

I don't think there's many myths surrounding West End Watches (they're just not popular enough), but one enduring one is that they were routinely issued by the military. This just does not seem to be the case.  I've only seen a few watches that were verifiably military issue.

The "broad arrow" printed on the backs of many West End cases is not the British government owners mark, it simply tells you how to align the caseback when you're replacing it.

There is plenty of circumstantial evidence to support this:

• To have just a broad arrow on it, it would need to have been issued by the British government (as opposed to a colonial government). Had the British government bought and issued any quantity of West End watches we would know about it. Military watch collectors are nerdier than your average watch nerd and it would have been thoroughly documented (though I recently read that they bought 50,000). 
• Military issued equipment also has an identifiable issue number stamped on it along with various other arcane glyphs. 99% of West End Watches don't have this.
• The "Broad arrow" actually isn't very broad. This is a broad arrow.
• If any government were to have issued West End watches it would have been the Indian Colonial government (as the company's target market) and their issue stamp was not just a broad arrow but the arrow above an "I" (for India presumably) and then flanked by various letters depending on the issuing service. The few actual issued watches I've seen had these markings and are of around 1920s/30s vintage. 

However, around WWII the Indian Civil Service does appear to have issued many wristwatches. At least, that is the accepted explanation for the CS(I) stamp routinely found inside certain styles of watch, usually on the dustcover or inside the caseback.

I say "accepted explanation" as I can find no source for this other than a note by Weselowski. He also denotes "W1234 S" as being a "Wristwatch. Service pattern mark", which all issued watches do have, but it's also a mark you find on lots of WE pieces without government issue marks and in many different letter permutations. So I think perhaps the general format is shared with some kind of importers stamp or similar perhaps.

The one thing the issued watches do have in common is that the issue marks are engraved by hand on the case rather than stamped. I also notice that the majority are Secundus models.