The game is on one console, so real time across runs should be near identical. I don't think there'll be a need for 'time without loads'.
For campaign runs, the timer should start on pressing A to 'Finish Briefing'. Starting it on selecting a profile doesn't seem right since it doesn't actually start the game, and you can still back out and the profile will remain blank. Starting it on the beginning of the mission would be inconsistent, since you'd have to use your reaction time to start the timer. Starting from a mission restart would be manipulating the loading.
For individual missions, the timer should start on the first black screen after loading the mission (in contrary to above). Missions are more likely to have closer times against each other, so accuracy is important.
Both should end when the (last) mission has fully faded to black at the end.
I forgot to mention it before, but there's also game time on the profile screen. I'm unsure how it's calculated, whether it's total time played or a 'sum of best' for each mission. If it's the former it can be easy calculated by subtracting the change, but if it's the latter it might not be useable. Starting runs on a completed file wouldn't work as it's only keeping the best times, so if you don't beat that time on a mission it won't change. Even if it was ran on a new file, restarts wouldn't be accounted for. I'll investigate this sometime.