To play both sides, open one side, say the British, make your moves and save - but do not click to resolve the turn. Then exit. Go to load games, select the other side, say the American, then make your moves and save. Now execute the turn.
You can tell if a side has a saved turn by the color of the link in the load game screen. Highlighted in red is a save has been made.
You can start the turn as either side and then finish with the other side.
It is a bit quicker than using the PBEM routine, but similar. I learned to do this because I was so used to playing both side on board games.
I forgot to add, this only works with a new game. If you have started a solo game against AI the opposing side disappears.