The required screening is probably due to an existing medical conditon in the family and may effect mortality as opposed to specified illness cover. Another insurer may take a differing view, although you would be required to disclose that you had applied for the present cover, and issue without screening. It may be quicker to arrange the appointment for the screening which will also put your mind at rest, so you can arrange the cover. Once again, someone who knows the facts could and should be able to help you a great deal.