SLIP AND TURN
Agreed. If indeed the separation was 500' then clearly something went wrong with one of the aircraft. As I am sure you are aware the TCAS doesn't take account of whether the aircraft are turning. Each calculation is based on rates of descent in protected areas from each aircraft. The collision threat is re-assessed irrespective of direction, So no more calculations would be made than if both were in a straight line.
Also both aircraft are on different frequencies which doesn't help. The only answer (short of moving one of the airports) is to ensure that rates of climb and descent are not excessive. This was part of my line training, even for my current airline that doesn't fly to City. In busy airspace we should reduce the climb rate to not more than 1500'/min when within 2000' of assigned level. Not written down, just good airmanshap.
Now RAs with military fast jet having a bit of fun in Pennine Radar, thats another thing.