Which bin for nail varnish and remover?
Usual answer in the UK
The tip Household waste recycling centre (the tip)
Nail varnish and remover are flammable. Take part-full bottles to the tip. Empty, dry bottles can go in general waste.
Also possible: General waste bin.
Does every council do this?
Most councils follow the answer above, but bins and sorting rules are set locally, and a few councils still do things differently. If your bin has a sticker, or your council sent a leaflet, follow that. Enter your postcode to go straight to your council's guidance.