I am afraid that Whitgift do not send out no thanks letters until very late in the process, if at all. But I would certainly think that if you don't have any post this week, you would be well within your rights to give them another ring at the start of next week.
I think that the Whitgift school places will be absolutely determined by end of their summer term (so 1st week of July), so not as late as September. Should any unexpected gaps appear after then, they will be dealt with through the 12+ process.
The three main rounds of movement will be a) when the other independent results come out (on the 13th of Feb), b) when state school offers come out (3 March) and c) when the first round of state school waiting lists get reallocated (end of March/early April).
For those of us waiting on co-ed schools the process has already started as many of the girls schools have or are sending out offers before the boys get theirs. Royal Russell, which is coed, have sent out their girls offers ahead of the reserve date (which was so oversubscribed they've gone for a second reserve date!). So some girls schools are already re-offering places and scholarships, before the boys know where they are ranked yet.
Whilst it probably sounds as if there is tons of running around, it is generally only done by a few families who fall on the boundaries of either getting a place or not, or getting an award or not. There will be plenty of applicants, indeed the vast majority, who will just return the paperwork with a sigh of relief.