So where does it go now, by the roadside perhaps?....

Yes, probably. IMO councils are developing a "not our problem" outlook.
At least Cambridgeshire has an alternative route for asbestos (with a
charge).

I now live in West Norfolk - who in spite of charging rather more
council tax, provide a miserably poor refuse service compared to any
other LA I've encountered - that must be a factor leading to
fly-tipping.