IAD local controller failed to provide separation to a go around and departure due to communications error and workload; resulted in a critical ground conflict.

Narrative

Aircraft Y was LUAW (Line Up and Wait) on Runway 19C with Aircraft X on final for Runway 19C. While I was waiting for the previous arrival to exit I gave ground permission to taxi Aircraft Z through the Runway 19C approach. Ground told Aircraft Z to do so without delay; but Aircraft Z took longer than anticipated. At this point I had to send Aircraft X around due to the traffic on the runway and issued a climb to 3000. After coordinating with the approach controller I inadvertently issued Aircraft A; who was on final for Runway 19R; fly runway heading climb maintain 3000. I gave these instructions to Aircraft A erroneously instead of Aircraft X. I then went back and issued Aircraft X runway heading and climb maintain 3000 and gave Aircraft X the approach frequency. Then I cleared Aircraft Y for takeoff on Runway 19C giving the pilot Runway heading as his initial heading. Aircraft A then asked what heading he should be on. Realizing he was going around I issued a diverging heading of 270 from Aircraft X and Aircraft Y. I maintained visual separation of Aircraft Y until appropriate separation between Aircraft Y and Aircraft X was established then switched Aircraft Y to departure. Aircraft X and Aircraft A were resequenced and landed later.Opening local 2 during this arrival bank to handle the departures would have alleviated my workload. Also pushing the departures to Runway 19L in the event local 2 couldn't be open would have alleviated my workload and we would have had more options with go arounds.