Your life sounds pretty similar to mine. I started losing weight before I had my son, but gained 50 back with pregnancy so since having him I've lost more weight than I lost before having him so yes, it is very very doable. That said it is a lot of work, but you have to want it. So here is how I do it.
The "diet"
I calorie count religiously. Every night while cooking dinner I prepare my breakfast, lunch and snacks for the next day and I prepare my son's lunch for the next day. To make this easier, lunch is usually leftover dinner and for breakfast I usually make a large batch of low cal low fat high protien super muffings on Saturday and keep them in the fridge. So the there is not much time to preparing all that. I use the LoseIt app on my phone and this take maybe 5 mins everyday to log in my cals. I also preplan my meals for the week on Friday or Saturday before I go to the grocery store so that I know what to buy and there is no thinking during the week.
The workout
I get up at 4:30AM and go for a run, usually 3-4 miles. Come home shower and then take my son to daycare. Now because getting up at 4:30AM SUCKS I take Tuesdays off and go for a run on Sundays with my son in a jogging stroller.
For you I don't know how old your son is so I don't know if a jogging stroller is an option, but if it is you could got for a quick jog with him in the evenings. Otherwise you're in my boat and you're are just going to have to suck it up and get up early. Showering at the gym is totally and option (just bring your flip flops
) I used to do this, but I don't have a memebership anymore and since I have to come back home to take my son to daycare it takes up too much time. So anyway, you can do this, but you're going to have to make some tough choices and sacrifies, but it's worth it. Good luck!