I pop what I can in the toilet then double bag and put in bin in utility - the bin will be emptied after lunch if all nap or at the end of the day.
Inspector followed me upstairs for nappy change and never said anything more as DH picked up nappy in bag from my drop off place on the landing bookshelf - I then went on to play with LO's in the den.

I sometimes think we over think things- it's actually ok to put things in bins indoors as long as they are emptied regularly and not accessible to children.