Widebody First Officer reports TCAS event with a departing UAV. Departure Control frequency seemed saturated.

Narrative

Flight departed normally and was given turn to ZZZ VOR after contact with Departure. We were given a climb to 13000 FT I believe and advised of traffic in the 11-12 o'clock position and rather close range- probably less than 5 miles. The autopilot was engaged at this time. It is hard to keep the big picture because we had a second TCAS RA again this morning. So much excitement in 24 hours! I believe there were scattered clouds; we looked visually for the traffic and never saw it. TCAS display gave readout of +100 FT. Hundred is correct. Departure Control was very busy; bordering on saturated. A warning klaxon could be heard during radio transmissions from ATC; which added to a feeling of urgency and alert. Hmmm; what could be happening next we thought to ourselves. I (pilot monitoring); requested verification of climb clearance altitude and we immediately got a TCAS RA 'descend now.' We were in a shallow climb to the new assigned altitude. Captain (pilot flying) assumed the full manual position and pushed the nose over without hesitation. Estimate descent was 500 FT. TCAS cleared the conflict; we saw the traffic -- a UAV working in the area; I reconfirmed climb clearance as well as notified Departure. We had a full TCAS RA commanded descent. 'Roger' was the reply. They were still very busy. We completed the climb clearance; were given a frequency change; I acknowledged it and again advised we had a TCAS RA just to be clear on the tapes due to the congestion.