Unless there is some local outbreak of infection - unlikely, since the nursery would be closed - nappies are not clinical waste. This is a LA matter, as these are trade wastes. Unless the nursely is very small, ie, little more than a child minder working from home, most enforcement authories would expect to see some evidence of waste management arrangements, collections, receipts etc. And if the wastes are visible from the curtiledge, and even more so if they accessible, there will be questions about standards, visual amenity, prevention of nuisance etc, with the likelihood of immediate service of notice.

I once asked the local council about a similar issue, they told me that the soiled nappies would be produced no matter where the babies/children were located and they would end up at the same waste reception plant anyway, therefore just put them in the ordinary, non recyclable bin for normal collection. I can understand there being a problem if not collected often enough to remove them all but they are not hazardous waste.