Just adding this analysis that I saw on Twitter. Not quite sure how interpret it but it's saying 10k ft or separation.
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I got 718 ft at the point of closest approach (PCA), but who's counting. The AAL jet started a go around fairly early, but then lingered around 1600-1800 ft for some time on the runway heading. Meanwhile, the DAL jet was climbing towards them like a normal takeoff, I presume. There was about 627 ft of vertical distance and 350 ft of lateral distance between them at PCA (15:52:52Z). By the time the aircraft were co-altitude, there was more than 10k ft of lateral separation.
Generally, in civil aviation, mid-air collisions (and mishaps involving wake vortices) are avoided by keeping aircraft positively apart.
One way to do this is to assign them different altitudes: if one aircraft flies at 5000 ft and the other at 6000 ft, there is no way they can collide (if they use the same altitude standard). The difference in assigned altitude is called "vertical separation", and it would be 1000 ft. in this example.
Another way to keep aircraft separated is to ensure that they're in different locations over the ground. If their tracks are far enough apart and don't intersect, they can't collide either. This is "lateral separation".
When a bunch of aircraft come in to land, they're often lined up on the same track, and changing altitude, so then the distance between them is used to keep them apart (2.5 nm at minimum, often more).
If you have at least one type of separation ensured, then that's safe.
In this case, as the incoming aircraft executed the go-around, it was flying above the runway, so on the same track as the aircraft taking off. But in this phase, the minimal altitude difference was 627 ft. according to the tweet you cited, so the emergency standard of 500 ft. of vertical separation was not breached: the aircraft were safe.
After that, one aircraft broke left, so now they had lateral separation since they were on diverging tracks, and could not collide: the aircraft were safe.
The 10,000 ft at equal altitude mentioned in the tweet do not matter from a safety perspective, since it was more important that the aircraft were flying away from each other at the time; 10,000 ft wouldn't be enough to ensure separation anyway. It's just an interesting data point.
So, despite the spectacular police video, there was never a risk of a mid-air collision. But there could well have been a risk of a runway collision, had the incoming aircraft attempted to land while the other aircraft was still commencing its take-off.
As it turned out, the departing aircraft would probably have taken off before the incoming aircraft, but the incoming aircraft would have had to contend with wake turbulence from the departing aircraft. Wake turbulence is like a horizontal whirlwind behind an aircraft that is the stronger the heavier the aircraft is, and it can affect aircraft flying too closely behind. Aborting the approach and going around ensured a safe landing for the incoming aircraft.
Exceptions to separation rules can happen when aircraft are aware of each other, e.g. the spectacular approach to LAX when two airliners are coming in to land on the parallel runways at the same time, or in military formation flying.
Airliners are equipped with a TCAS system that tracks other aircraft in the air, and commands pilots to take evasive action when it detects an immediate collision risk.
