What can cause output voltage of DC-DC Switching regulator(AP6502) to drop when load is applied?
The ground system you are using is a train wreck! - I've highlighted in pink where it is on the top side of the board: -



And on the bottom side: -



Maybe it's my eyesight but I can't see the all-important C2 (power decoupler to the chip) either.

You really need very thick tracks/ground plane/sensible star-pointing of grounds for this to work as you might expect it to. For a start the return path of C5 should be back to the ground on the chip - there are rather large pulses of current flowing down those tracks and they should be as short as possible. Instead you appear to have the bottom end of C5 snaking around some very sensitive areas of the feedback components.

You've deleted the circuit so I can't tell you which cct references.