To be honest, I wouldn't worry if I was you. Assuming you're in year 11, you won't be applying to uni for another 2 years, and that's plenty of time to find placements 🙂 I applied for veterinary medicine and had the same problem as you; I didn't end up meeting my required work experience hours until the summer before I applied. My best advice would be to write a cover letter and/or a CV and just email absolutely everywhere relevant. If everyone says no, try again a few months later - keep contacting them, and eventually someone will say yes! Try to keep an open mind too, obviously if you're doing a course like medicine then experience in a hospital is really helpful (and clinical placements such as that may be an entry requirement for some unis), but part time jobs/ volunteering in places like cafes or nursing homes etc. is often accepted as valid work experience too, it just depends where you apply (so might be best to check with some unis to see what they accept generally).
Good luck 🙂