"What happens when an aircraft is taking off on the western end at the same time as an aircraft landing on the eastern end gets into trouble and needs to do a "go around"?"
they are several kms apart - if they are both still moving the one in trouble executes a standard Missed Approach and the other is instructed to turn the other way if there is any problem
The only issue is if the one in front rejects a takeoff below V1 - but if the runway is long enough he'll be in Slough when he stops while the other guy will be turning off the runway to the terminal before the spot the western aircraft joined the runway before starting his run