When I try to upload, the RX led on the USB2Serial board flickers for some time, but I think the Atmega is not reset (at least the led on pin 13 does not blink).
Unrelated note: You really should not have AREF connected directly to Vcc.
If you try to upload a sketch does the D13 LED blink twice the same as when the reset button is pressed?
Oh, good to know! Hmm, the tutorial I linked had the AREF connected to Vcc... I guess it should remain unconnected?
Problem solved! The culprit was a blown capacitor. Thanks to all for the tips that helped me to get on the right track.