It seems like the tap-to-wake feature on my Lumia 1020 is not very reliable.
Is there any way to make it work more consistently?
You may have set your touch sensitivity to "High".
For some reason, this actually seems to make the phone less responsive to tap-to-wake.
Try setting it to "Normal": SETTINGS SYSTEM DISPLAY+TOUCH TOUCH TOUCH SENSITIVITY
Other than that, keep in mind that the two taps must be:
With "Normal" sensitivity and a little practice, you should be able to tap your phone awake every time.