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?
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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.