"Monitor stays black" lag?
That Lenovo TB3 box has 2 DP (display port) and one HDMI port:



( picture from here )

I simply switched ports, from DP2 to DP1, and using the first port, there is (almost) no more lag any more!

And as reference for future readers: if you intend to connect two monitors to this Lenovo TB3 dock: the second monitor has to be connected using the USB-C port on the front of the box. When you plug two monitors on the "back" of the dock, they will show the same content (like only "mirroring" is possible). 

Update: 

I figured a bit later that yes, switching the port did help, but only to a certain degree bit. A slightly better "solution" boils down to this:


when I come back, and both displays are black, I click my mouse once
I count to five, I hit ESC on the keyboard
I count to five again, and click my mouse again


What that does is: (almost) reliably get met to "both displays awake at the same time". 

( I basically noticed that clicking the mouse will wake up display 1 quickly, and when putting it back to sleep with ESC, the next mouse click will wake up both displays ) 

Another update:

Since upgrading to Catalina, no more workarounds required. display 2 still has a lag, but I simply enter my password, display 1 turns on, display 2 follows 2, 3 seconds later. Reliably. At least for now.