It is normal to have to wait until you are 16 or 17 to do clinical area work experience in a hospital due to health and safety concerns. If you are younger some hospitals will let you do work experience only within the administration departments, so you would be shadowing secretaries etc, but would not be allowed with doctors or nurses or near wards or patients.
In the meantime try looking for volunteering opportunities at care homes or work experience at GP practices (some care homes and GP practices will accept under 16s, but not all). Even if all you are doing is reading to the elderly it is good work experience for your personal statement.