Traffic shouldn't be bad that time in the morning. I've left home (Kensington) at 5am for a 7am departure several times in the past couple of years (during the skiing season ) and it's taken the minicab about an hour, irrespective of whether they go due south or whether they take the (better IMHO) route of A3-Tolworth-M25 junction 8(?)-M23. For an 07.40 I'd leave home at 05.30 to be safe because it's just that little bit later and people start waking up etc. Add 15 minutes or so for the fact you're starting in Piccadilly and an 05.15 start should be fine, unless you're particularly nervous.

GEX to North Terminal is a total pain while the shuttle train isn't working, and I understand it can add quite a bit to your journey time.