I've been struggling with health/weight myself and stressful work + personal life. So I feel for you, what you are going through takes its toll on a person. It's also great that you want to change this, because you can.
The most important thing I realized is that there are two factors at play: mental health, and physical health, and they go hand in hand. If I have a stressful week, I am more likely to not exercise. If I feel bad physically, e.g long nights staring at the screen, no movement, I am more likely to get stressed at problems at work. It's a vicious spiral.
The good thing about that realization is that feeling better physically also helps make taking things at work easier, and on the other hand, being able to distance myself mentally from work gives me the peace of mind to go and do some exercise. So any way you approach this, the result is the same: healthier body and mind.
With that said, what I do practically is the following:
I log off at the appropriate times, make sure to not have too many tasks at the same time (best is only 1 at a time), and keep reminding myself that work is never going to finish, and I can always continue tomorrow. I also try mindful breathing whenever I feel like it, and some guided meditation. Also, I stopped joining work meetings where I think the information is not beneficial for me or the topic is not interesting. And I listen to chill music with OK-ish noise cancelling headphones.
On the physical stuff, I stretch, make some light bodyweight exercises, and hit a boxing bag once or twice a week. And I sometimes go bouldering (despite sucking at it). I also try to have at least an hour of a walk daily, whether it's going to the store for groceries, or just walking around and looking at trees or that kind of stuff.
These strategies don't always work for me, I don't think there is a silver bullet in these kinds of matters. There are days where my shoulders are stiff from the work pressure, my mind is spent, and I just want to eat junk food and watch Netflix, and that's okay. Following a plan too strictly, at least for me, is a sure way to not do anything or get stressed if I miss a day of exercise. Let yourself not be perfect, cause nobody is.