The UK issues these watches

http://www.cwcwatch.com/collections/general-service-watches/...

I only ever saw them when checking stores - Q would never actually let you take it out and use it!

Maybe you were 'issued' them as well, but like ours they're never let out of the stores?

This could be a hold over from habits established before cell phones became common. Back before, say, the '90s, most people in the US (and I assume in most other first world countries) from their teens onward had a watch.

Perhaps militaries assumed that everyone coming in would already have a watch.

Furthermore, watches were often of great sentimental value, and many people would have viewed the military telling them what watch to wear as being as absurd as, say, issuing a standard military wedding ring to married soldiers to replace their civilian wedding ring.

As long as a soldier's watch was accurate and reliable enough for military needs, they probably saw no reason to not let the soldier use his own watch.