I think there are socio-economic factors at play, yes - these are well-documented and range from being time-poor, resource-poor and the availability, accessibility and affordability of fresh food, and of course 70-80 % of your physical build is genetic, some health conditions and medications mean you put on weight more easier and so on.
But in your situation, it is not socio-economic factors because, as you say, your husband works out almost daily and looking good. It is the fundamental inequality in your home. Your DS’s health issues have been made your problem to solve by your husband.
Why on earth was taking him out of nursery the proposed solution by your DH? That was bound to impact on you. You have a toddler with sleep issues and your DH is at the gym every night. By the way I fully sympathise, my DS had ear infections, he would not settle at all in nursery or go to sleep before 10 or 11pm. It drove me to distraction and to be honest, my marriage ended because it was all put on my lap, plus my husband’s life did not change one iota, and my husband was basically a control freak about how things should be.
If you do not have equality in the home, you will not have equality outside it.
Things I did:
Changed nursery to one with lots of outside space. The previous nursery was an old town house with wooden floors and high ceilings - way too noisy for DS. He thrived with the outside space and more flexible routine. So, just because your DS disliked one childcare provider, he will not dislike them all. Starting looking for alternatives. Try to figure out what exactly DS disliked about the old nursery.
Exercise - we aim for 60 minutes a day, whether that is swimming, cycling, walking, park time. You can build a routine around it. This is possible even if you work. I cannot afford the gym and I could not leave DC to go, so this involves them. I learnt to dive and can swim 20 - 30 lengths in the half hour DS has his swimming lesson now, for example.
Ask your health visitor if there are any healthy lifestyle programmes for families in your area. I have done two with DC and they are excellent. Paradoxically, also helped very slim DD who was prone to anaemia. Take whatever support you can get as you are in a difficult position.
None of this will be solved overnight, but it is about making small changes to a healthier lifestyle.
Ideally, your DH would recognise that leisure time (for the gym) should be split equally, and you would both look for an alternative childcare situation so you can get back to study. If that does not happen, then you need to break down the issues into manageable chunks and seek what support you can elsewhere.
(And then when you are a hot shot lawyer, you can draw up your own divorce papers
)