I think you are taking on a lot, too much but your marriage dictates that you get yourself into a more employable place.
If your DC2 is old enough to be wrecking the living room then I'd suggest that you should seriously consider ending the breastfeeding. Certainly wean him/her off any overnight feeds as a matter of priority as it sounds like you need all the good quality sleep you can get.
If you can get more sleep, and get to bed with lights out by 10pm, you could consider getting up at 5.30 / 6 and getting some study in before the kids are up and about. Easier said than done if they are early risers though.
Make a plan for the housework - it simply has to get done during the 45 hours a week you spend with your kids even if that means less "activity time" for them. Put together a rota for yourself so that you dont get to the end of the day and think "shit, I was supposed to park the kids in front of cbeebies for 20 mins and hoover like a madwoman. If you have a travel cot, break it out and use it as a playpen if necessary to contain the mess/risk while you run around.
Trial the new job for a month and keep your eyes peeled. What opportunities are there to take advantage of internal training? Are there admin jobs that everyone hates doing that you could take on on a Saturday. Trick is to show willing and get yourself into a stronger position internally so it's not just about increasing your hours [which there is little point to if it doesn't cover your childcare costs]
Does your family income exclude DC2 from any free hours?
Your mum / husband sound deeply unsupportive. Is she retired? Next time she has a go, can you ask her to come for a week to look after the kids so you can head to the library and cram ahead of your exams. Get her to put her money where her mouth is or STFU. Same goes for your husband. He presumably gets at least 20 days a year off plus bank holidays. If he has three degrees, he presumably has a good job and probably more than the statutory holiday allowance. Can you build in at least a study day a month? He takes a day off or two half days when there is no bank holiday; and you disappear to library or he takes the kids out all day.
The running is for you - if you can't run with a buggy [hardly "me time" is it] then I'd suggest that you eat with the kids and head out as soon as they are in bed and DH is slaving over a hot iron.
Your new salary doesn't cover childcare but it has to be at least minimum wage - if DC1 is in nursery a few/5 mornings a week, can you pay someone minimum wage to babysit DC2/take him/her to a toddler group for at least 2 of those mornings? Nursery workers are paid peanuts as far as I can figure out, and are often on shifts.