The gym isn't a no no - you just have to be careful. It would be a wise idea to speak to an exercise professional (no, not your CDC unless he or she happens to be a personal trainer as well - believe me, there's a lot more to exercise prescription than what's included in the Cambridge training) and get a plan drawn up. It's perfectly possible to do moderate exercise whilst on CD, including cardio - key word being 'moderate'. Vigorous exercise is a bad idea, because your body can't keep up with the demands you're making on it. However, one person's 'gentle' is another person's 'moderate' is another person's 'vigorous' so use that gym membership, go and ask for a fitness assessment and a training plan that's appropriate for your nutritional intake.
But to answer the original question, you shouldn't be doing enough exercise on CD to have a noticeable effect on your weight loss - if you are, you need to be eating more.