I believe your local council - district or borough - has an obligation to dispose safely of medical sharps waste.
Certainly my local council, Arun DC in West Sussex, provides an efficient service of collecting from my doorstep at 07:00 (on the dot!), a sealed and full bin on a set day of the week, and leaving an empty equivalent for ongoing use.
A thorough search of your local council's website should bring this to light - try 'Cleansing', 'Waste Disposal', 'sharps', whatever comes to mind. It would be a particularly obtuse council that wouldn't do this for you. :thumbdown:
Good luck. :clap: