I'm not a mom, but I work FT and go to university FT, so my time is probably just as limited as yours. I don't do a whole lot of the primal exercises (I do body weight exercises [love love love push-ups] occasionally run stairs when I need to clear my head). You don't have to set aside time for 3+ hours of hiking, but doing smaller things like parking as far away as possible from your destination (extra bonuses: no more dings from people's car doors/shopping carts) or taking the stairs will definitely add up.
When I'm around my niece, I have NO problem using her for strength training. Tossing (and catching) a kid for 10-15 minutes will work your arm and leg muscles, laying on the floor and doing 'airplane' with her (legs and arms, depending on how high my niece is comfortable with that day). Kids even work kinda well as kettlebell exercises with the swinging motion.
I'm sure most day-to-day chores if done repetitively (raking, sweeping, lifting groceries, et cetera) are great exercises in and of themselves.
Warning: I'm the most laid-back primal gal you'll ever meet.