Yes, naps actually help you recover, especially if you have a lack of sleep. It won't hurt you. There is no such thing as a "bad nap". Since you're just doing basic bodyweight exercises, stretching, jogging, all of which are pretty low intensity..I'd say the lack of sleep will not affect you. These help keep you in shape, but unless you're working out with weights to gain size, strength, or endurance, you do not need to worry. Otherwise sleep is a huge factor in building size and strength, alongside diet and workout. I would advise to go to bed earlier and get 8 hours of sleep or even take shifts, so that your wife can watch while you go to bed at night so you can catch up yourself. breaking up your 8 hour sleep pattern will hurt your size or strength gains but if you have to you definitely need to try to get naps.
Again, your workout is not very demanding in terms of your body's needs so it should not affect or be affected by lack of sleep.Just rework your schedules to get 8 hours of sleep for your own sake of not being sleepy. Try jogging during lunch break or after work and doing stretches before bed? It sounds like you do not need a gym so you should be all set.