Hi XmaX,
Bag Drop seems to only be open from two hours before departure. I go through LCY a lot and I've seen many passenger turned away for being too early. It's a small airport, so i suppose they really don't have anywhere for the bags to "wait" after bag drop.
Here's the detail confirming that on the LCY website:
https://www.londoncityairport.com/tr.../page/check-in