Goodwood runway conflict?
 

At Goodwood yesterday, the classic cars were displayed on the racetrack around the end of the active runway: the regular noise of an aero engine was thus normal.
While looking at the cars, I became aware that I could actually hear two separate engines, both at high power. One was an aircraft on the runway, that started its take-off roll: the other was an aircraft about to land on the same runway, that had initiated a go-around at no more than 50 ft.
The a/c on the ground continued with its takeoff and became airborne: it climbed towards the a/c on its go-around and thus occupied the same airspace and track, and they must have come very, very close to each other. From a position now directly behind I couldn't tell which one if either was in front, and thus possibly had visual on the other.
Does this sort of thing happen very often (the mind boggles) - and what should have then take place to get the a/c as far apart as possible within the shortest possible time?

The aircraft performing a go-around should track on the dead-side of the runway (eg to the right if using left hand circuit) to keep aircraft departing in sight.
The need to perform a go-around is not uncommon and is usually a sign of good airmanship.

G-XLTG

Depending on your position, they could have looked closer than they were due to foreshortening (think of the Red Arrows manoeuvre where they appear to almost hit each other in the synchro pair - they don't, it's an illusion)

Yes, I'm aware of that, also that a 'go-around' in itself is just a fact of life and nothing to get excited about.
My point though is that I assume both aircraft must have been under some sort of control: who then directed two aircraft to simultaneously use the same runway for different purposes, therefore relying on uncommanded and last minute evasive action on the part of one of them to avoid a collision? It seems like taking an awful lot for granted. Or were they just 'I Follow Roads'?

It is an uncontrolled airfield therefore the separation of aircraft is the respective pilot in command's responsibility.

It's a FISO service so it will be "take off/land at your discretion" and it's up to the pilot whether to land or take off.

<<that had initiated a go-around at no more than 50 ft. >>

Nothing too exciting with light aircraft. I've seen a 747 go-around lower than that!