How many days after radiotherapy should chemotherapy be taken? If too soon, does the patient get really really sick?

It depends. There is no single answer. In early breast cancer the delay might be 4 weeks. In lung cancer, bladder cancer and rectal cancers the treatments are given together to get the best treatment effect and shrinkage of the cancer. When these treatments are given together the doses and schedules are often changed to reduce side effects. Most patient are able to take these treatments without excess problem.