Assuming the 1020 has the wireless charging case of course.
New Qi Wireless Dock Cradle Holder Charger for LG G3 Nexus 4 5 Lumia 920 BC406 | eBay
New Qi Wireless Dock Cradle Holder Charger for LG G3 Nexus 4 5 Lumia 920 BC406 | eBay
If I buy this charger, does the time continue to display on the phone screen while it's charging so I can use it as a clock in the middle of the night or does the time eventually go away?
It says it is a three coil charger and that it works in landscape mode as well as portrait mode so I would not think that is an issue.The only issue you may have is that the qi on the 1020 is slightly lower on the phone than on the 920, due to the camera.
Worst case scenario is that you would have to put a small block under the phone to elevate it slightly.
Chances are it will work fine as is though.