When I was under Falkirk District, I got sharps bins from the doctor themselves. In Ashfield District, I got them on prescription and the council picked them up when I was done with them which I always forgot so had about 8 piled up at once, they didn't like collecting 8 at once for some reason. Down here in Vale of the White Horse, I have to go into the doctor's surgery and swap my used bin over for an empty one. Mine is massive because I get so fed up traipsing down there all the time, however, you can get little ones.
I don't know why they can't just standardise it so that every place does the same. I didn't get a sharps bin until about 2005 because I was told just use old drinks cans and put a bit of sellotape over the opening then throw in the normal bin.