Quite often at Luton you can see a flight at the distant end of Rwy26 with an inbound aircraft visible on the approach. It all works out generally well but, when I was there recently a departing Wizz Air flight had been cleared to go but didn't move immediately. Meanwhile a Ryanair flight was getting closer and closer. Eventually, it was so close I began to wonder what would happen now. As I was thinking that, it all happened at once. The Ryanair flight changed his mind and began a missed appraoch. At the time, Planefinder was showing this one at 350ft. As he began the procedure, he was directly above the Wizz Air flight which decided it was time to go so there were 2 aircraft travelling in the same direction. 1 on the ground giving it max revs and another directly above him a couple of hundred ft above. Within a couple of seconds, the Wizz Air flight made a heck of a noise with the noisiest brakes I have ever heard. The Wiazz Air then trundled sedately to the 2nd junction, turned off and set off along the taxiway to join the back of the queue. Almost nonchalant! I wasn't listening to the radio but I bet there some stern words!

I've seen that with two 747s. The one going-around was given a heading to take it away from the track of the outbound which continued on its way. Job done.

We did it at Farnborough with 2 Antonov 124s!! The one lined up was given takeoff clearance as the landing one turned downwind and still he hadn't rolled when the landing one, which did a 7 mile final, was at 2nm!
After the airshow that year, the controller involved was posted to Heathrow to do Thames Radar (MB)

I've seen that with two 747s. The one going-around was given a heading to take it away from the track of the outbound which continued on its way. Job done.

I always wonder at the routing of GA's. e.g. STN RW22. The GA is towards BKY at 3000'. The Buzzard/CPT SID is towards BKY at 4000'. It has the potential of 2 a/c flying the same routing, with the GA a/c perhaps going faster than the takeoff a/c, and the lower a/c climbing through the GA. It then requires the Tower controller to separate them without necessarily being visual and not necessarily having the 'big picture' about all a/c in that part of the sky. And then Tower radio goes dead. It would not be a good day.
I just wonder why this potential conflict is designed into the procedure; and it's not only at STN.

Around 1978 I saw an Aztec line up and go on 28R at Heathrow with a Pan Am 747 practically over the fence.

The 747 went round, moving to the left of the runway (ie closer to the central area), presumably so the First Officer could see where the Aztec was. The Aztec stayed low, from the Queens Building Roof I didn't see it much after it lifted off.

Didn't have an airband radio in those days so I've no idea how the conversation between the three parties went; I imagine it may have been a little tense!