The club have done no analysis of start times/finish times/success ratios/etc beyond the annual report, examples of which may be found on the club's web site. I'm not sure what Brian (Covell) kept with regards to attempts rather than successes but since I took over his role three years ago it is something that I have thought of presenting. Such data would be highly anonymised and along the lines of: year: 2005; month: 5; start_time: 0000; direction: clockwise; leg 1: 233(mins); leg 2: 258; leg 3: 374; leg 4:315; leg 5: 192. (These are from my round) obviously if someone has done a fast time or is the only anticlockwise round of a year then they are going to be identifiable. It's on my "TODO list" so don't pester me for it
There are obviously certain caveats that need to be applied to any such data, namely there's little allowance for weather or ground conditions or how any individual reacts to those conditions, I suffer quite badly in conditions of high humidity for example but others might find that OK. Also just because someone registers an interest in starting at midnight doesn't mean that they do so: they may change it due to weather or support problems. This problem is most apparent in those who attempt the round but don't succeed since I can't update my data to show this, it's likely to be a minor problem though so the general relationships should hold.
My current thoughts are that you should aim to have the runnable and tricky underfoot sections in the daylight, this means (for a clockwise round) getting to the summit of Blencathra at first light if you are aiming to do Hall's Fell or the parachute descent then you've got the descent to Threlkeld and all of Leg 2 and Leg 3 as well as most if not all of Leg 4 in daylight. This leads to a midnight start, a 1am start if you wish to be sure of daylight on Hall's Fell, maybe 11pm if you are thinking of using Doddick Fell then you are likely to get sunrise somewhere between Clough Head and Great Dodd.
I've tried a 1900 start and had an awful time (at night) on the Helvellyn ridge struggling with both fog and rain in even locating the path, stumbling over protruding stones that were nearly impossible to pick out. I think that you need to know how much the darkness affects your running speed, the Skiddaw leg is generally slow enough that darkness doesn't make that much difference especially since the route is increasingly obvious on the ground.