CM12.1 - My lock screen is not working at all!
Answering my own question here, so I can mark it as the solution. (I also posted this in the original question as EDIT #2).

I just noticed in my quick settings (i.e. the pull-down from the top of the screen) that there is now an icon to turn lock screen on/off. I guess this is a new addition that I failed to notice, and 99% sure this is what caused the "problem" -- which was really a feature. I must have accidentally tapped it off and then back on before even noticing it was there.