Go to settings>device>sounds and notifications >click while locked and choose show all content. This should fix it or you can individually choose application notifications. Hope this helps...
I agree this is helping people, but I don't think that option is disabled by default on a lot of phones. As the OP has mentioned, the current state of affairs with lock screen widgets are bad. I have a Moto X 2nd gen on 5.0, and a GS4 on 4.4.4. Guess which one I use for music in the car? GS4.
I, like many others here, will use this thread to rant a bit:
I use Spotify. The lockscreen widget on Android L is a joke. Pause/play, Skip forward, and ..... Close button? The damn thing is so small and the close button is right by the skip button. Guess how many times I've ended up just closing the app instead of skipping forward. Sure I can "pull down" to expand but that requires an extra step and looking at the screen... more dexterity requirements. Even then it is still tiny and useless in the car.
While we're at it, I just want to say that while there are a few good things about it, the notification concept is still flawed in Lollipop. Pull down to expand, but also... pull down to pull down? And sometimes when I have a lot of notifications I'm doing a lot of scrolling just to get to what I want, or to get to the quick settings. Sorry, two finger scroll is hard with one-handed use. And why can't I close apps by swiping to the side like I can dismiss notifications? I don't want an X button. Stop rubber banding and let me freaking close you if I want damnit.
Also while we're at it, the sound/vibrate/dnd in L is flawed too. If I'm Chromecasting I can only ever ever control volume on the Chromecast (um, wouldn't I just use TV remote in most cases?) I can't turn off my ringer or lower game volume if I want to play a game at the same time without using a separate 3rd party widget on the homescreen. Or if I'm only in a game I can't turn off notification sounds because it controls game/media volume unless I leave game and wait like 30 seconds. It's just not right I tell you!
Rant over. Google, please make it better.