Hi, mum. I can imagine how frustrating it might be. A short answer to your question is, you need to increase your activity as you do not have enough time to exercise.
Now for the long answer.
1) Weight loss after 25. It becomes difficult to lose weight after you turn 25 as your metabolism settles down. Even you end up taking a desk job, and the activity level reduces. Thirdly, if you continue to eat as you used to as a 25-yo, you end up gaining weight.
2) Raising three kids is stressful. So, tendency to stress-eat or sleep less due to stress increases. Both lead to increase in weight.
Solution
1) Be careful about heavy exercising. If you are not getting enough sleep, hitting the gym and burning all the fat might cause more harm than good. It puts a pressure on your heart, something you do not want to happen.
2) Increase your activity. If you are pressed for time, try and modify a few activities that you do. Climb at least 3 floors at least 4 times a day. Walk while you can, and avoid taxis/cars wherever you can. Even a walk to and fro from the MRT is a good exercise.
3) Use the baby to your advantage. Get a comfortable babywear and carry the baby instead of using a stroller. The constant weight of the baby will strengthen your back.
4) Measure your activity. If you use a smartphone, you will find many apps meant for measuring the steps. Like the PM suggested, aim for 10k plus steps each day.
5) Watch your diet. Don't go for any fancy diet as your baby needs all the nutrition he/she can get from you, if you are breastfeeding. Instead, aim for a balanced diet. Carbs, proteins, fats, all have a role to play. If you cut down one and increase other, it will end up harming your body in the long run. You can consult a professional diet expert for your nutritional needs.
6) Watch the intake of beverages and sugar. You do not want the additional calories. Even sugar-free options lead to obesity in the long run.
Mum, get a good support group as it is important for your own good state of mind. I hope my answer helped.