Hello Sparkle:)
Quite newly a single mum here, although ex has the boys a few times a week. I work from home so I have more time, but these are the things that help.
I keep a spray of watered down bleach (in a spray bottle., cheap, don't need to buy expensive cleaners, also add lemon) in my bathrooms so when the boys are getting ready I have a quick wipe round a couple of times a week. Then I give a good clean about once a month. Quick bit of bleach down the toilet and a wipe and its done.
I have a slow cooker, it cooks enough for about 2 meals for all of us plus a lunch for me. Dead easy, and I can leave it on all day or overnight. Freeze them for quick meals later on.
Kids always put their plates away, and near the sink. They lay the table and tidy up too (5 and 7).
I have a diary, and a calender that organizes me otherwise I would be terrible.
I studied when the oldest was very small my MSc in nursing and worked full time, its really hard but you need to try and do say 1 hour a night study and not watch tv or surf the net, try being uber focused just for 1 hour after the kids are in bed. Don't try and cram loads into a short space of time, you get stressed and then start to lose focus.
About the gym: is there a kids club there? Could you take the kids once or twice a week after school? I pick my eldest up from school twice a week and take them to the gym with a pack up (they have hot school dinners), they have a small room where they can play or they can go outside and play football (its quite relaxed where I live overseas so it might not help you at all).
If you plan and cook your meals in advance or spend a couple of hours meal prepping a week, the weight will take care of itself. In the gym don't spend hours doing cardio. 45 minutes resistance training is enough. Use the weights, trust me on this you wont get bulky. The more muscle you gain, the more you burn calories and your body fat will drop. Don't restricy yourself too much, and make sure you have a good amount of protein in your diet. Fitness pal is good, but might take up too much time to monitor your macros and calories. But don't put too much pressure on yourself. Drink lots of water too :)
I admire you like I said before, its not easy studying as a nurse with no other things on, you are doing a great job!