As someone who has worked in a stressful job with little free time, I understand how hard it can be to stay in shape. While I am in a better situation with regular weight and cardio workouts, there was a time I saw my weight go up and up. I did finally take charge and started doing "mini-workouts".
Whether while brushing my teeth or during bathroom breaks at work, I would do pushups, squats, lunges, or calf raises. Body weight, of course. While this does nothing for your cardio health, I was able to increase my strenght and lose a bit (5 - 10 lb) of weight. I was up to 2 sets of 40 pushups twice a week. Squats and lunges were 50 at a time. I did 90 calf raises per set. I even managed to do some neutral grip pullups using stall doors. Each "workout" lasted at most 5 min on top of my bathroom break (people have smoking breaks, I have workout breaks). And i know, pushups on a bathroom floor is nasty, but there's soap and water nearby.
Point is, if you want to squeeze in workouts, you can. Even though I go to the gym now, I still take stairs going up at my workplace now. Sometimes I toss in some body weight squats while brushing my teeth. Watching TV? Crank out 20 pushups during commercials. If there's a will, there's a way.
Also, diet is actually the most important part of losing weight. Minimize sugar, salt, junk food. Soda? Drink diet. Coffee? Drink black no sugar, or at least milk and artificial sweetener. Need a snack? PB&J on one slice of whole wheat bread. So many ways to cut down.