Atcos involved didn't make the procedures they operate under (ie what runways in use at the 2 airfields simultaneously). Had a double GA and had to do some rapid thinking to resolve. We've all been in sudden and extremely stressful situations, no point in thinking about how you got there, just resolve situation and give out later.
Worst case scenarios should always be planned for, however management are not always willing to do this when they can run with their new procedures to their bosses and show how many more movements they can get with their bright idea and get themselves promoted. I don't work in the States and I don't know what the procedures in NY are, I'm just trying to make a point in general.
You can go to skyvector.com and check out the layout of KJFK and KLGA.
The correct course of action would have been to direct the 747 to maintain runway heading and climb to X,XXX. Normally, the 747 would have been directed to turn right to approximately 070 but that was precluded by the parallel runway go-around in progress.
Instead, the controller turned the 747 left to a heading of 250 which, except for Lady Luck, would have resulted in a mid-air.