Always after DS has gone to bed. I'm terrified of dropping the iron on him. Usually on a Tuesday. I sort the whole week's laundry into folding to put straight away (towels, nappies, underwear); stuff I need for the week, which must be done that night; and stuff I'd like to get done but don't have to (babygrows, outgrown clothes to put away, items we won't wear/use over the next week). The must do pile is always the smallest, which makes it easy to make a start and once I've started, I get a rhythm going and always do more than needed.
I prefer the radio to TV for ironing. Most of my clothes probably ought to go to the dry cleaner as they are typially very fitted dresses (I'm very hourglass and look dumpy in anything that isn't extremely shaped) so they have complicated pleats and darts. If I wasn't looking at the ironing board constantly, I'd iron in more creases than I take out. I tend to look for a documentary series on IPlayer radio with lots of episodes and just go through them.
Otherwise, if DH is home, he has instructions to watch something on the television that will neither interest nor annoy me, usually a sci-fi type thing, and I just let my mind wander.
All DH's work shirts do get done by the dry cleaner. When I think how much either of us would earn in the time it takes us to do one shirt, it looks like a bargain, especially compared to what they charge for a dress.
If your stuff isn't very fitted and cost isn't an issue, might a flat bed ironing press save you time?