Hi Richard
QE2 departed Southampton for the final time 'officially' at 19.15 hrs
Earlier, departure times varied - often from Southampton at about 17.00 hrs - or soon after.
When QE2 was away from home, the schedules varied, apparently depending on Port activities, availability of Pilots, Tugs, Shore-side crews etc etc
Often there were stop-overs overnight - sometimes twice in one voyage - e.g The Baltic (Copenhagen and St Petersberg), and late night departures - Liverpool, Fireworks etc
All 'booked' years in advance.
Times of departure were published in every one of the Cunard Brochures
For Soton - try the Port of Southampton Authority site
http://www.southamptonvts.co.uk/someone there may be able to give you more historical info
And the QE2 Archives in Glasgow might give you the time of both her launch, and 'Mini' Maiden Voyage
http://www.gla.ac.uk/media/media_71009_en.pdfLouis (Blue Bombay) will know more from his diary / records too!
Rosie