Well I do think its a bit rude of you Roffa old chap, as even though I am not an overworked controller with not enough meal breaks, I have had quite a few hot dinners whilst watching that piece of sky in the last few years
I still haven't studied the plates but all I can tell you is that when they LHR & LCY are using these western runways (which of course is frequently) and when there is no cloud obscuring my view or theirs, more often than not, the LHR traffic turns south around the eastern edge of CW and then turns roughly west around my gaff.
Sometimes it extends downwind and might then turn 135 degrees back toward me, and less frequently it might turn earlier down near Tower Bridge.
I am little surprised but not overly so when you say that barring unusual circumstances the traffic from each airport is handled autonomously.
I guess you must be a controller.
I can equally well tell you that the City departures yield a more consistent pattern with usually at least 90 degrees of the right turn out completed by the time they reach abeam Canary Wharf. Not always however. An hour ago while I was having another hot dinner I noticed what appeared to be a merry dance of City traffic extending on the runway heading well past CW then turned north while something heavy joining the LHR approach from the north made a kind of S on the LCY side of the departing traffic.
I am sure what you say about the published procedures is correct but I am just reporting what I see. I am sure that if videos of the traffic were overlaid Red-Bull-Air-Race style then the overlaps would be quite apparent. Not too many level overlaps we would hope, and no simultaneous ones, but the tracks certainly would all the time and that just leaves our 1000 feet to do all the work.
If Red Bull had left their cameras in place the other week covering the sky just next door to this bit then maybe you could be more sure ...
The difference between this corner and any other in the London TMA is rather more obvious than you seem to be suggesting Roffa. As I am sure you appreciate, they are up to a rate of around 2.5M per annum passengers out of LCY I think and in small aircraft that's a lot of movements, especially if they are sometimes only half full!