You could walk. Walking is excellent cardiovascular exercise. There are many members who will tell you how they lost 100+ pounds just by walking and watching what they ate. You can climb the stairs. Climbing stairs is also excellent cardiovascular exercise.
You could do squats and lunges. You could even do some strength training. Oddly enough, if your right arm is broken, you can still strength train with the left arm and still get benefit.
No, you won't be lopsided. Yes, the left will be slightly stronger than the right side for a short time. However, when you work one muscle, that pulls on another. So, yes, you will still benefit from strength training if you only work the one good side. You'd be surprized how quickly your broken arm will respond to the strength training once you're allowed to return to normal activity.
Here's the thing, we exercise to keep our bodies fit. If you're trying to lose weight, you need to be mindful of what you eat. Good nutrition is what takes the weight off and keeps it off. Exercise is what keeps our bodies fit and healthy. You can still lose weight if you have to modify your workouts because of your arm.
When it comes to weight loss, nutrition trumps exercise.