I've not had that problem, but, I suppose that, if you can find something in the rubbish that identifies the tipper, you can get them to pay for the removal. A long shot, I know, but that's your only redress.
I wouldn't think that the National Highway Agency would have to pay to clean up private property. If they did, then I might not bother putting my bins out...
"Now that the agency has concluded its investigations, arrangements are being made via Road Link for the safe removal and legal disposal of the fly-tipped waste from the A69 layby and the adjoining landowner's property, and we're in contact with the resident to arrange for this to cleared as quickly and effectively as possible."
So a non story then, as they are going to clear it anyway.
^^^ The Highways Agency will remove it but then send the clear-up bill to the property owner.
It's no difference to any other offence committed against one's property. (e.g. where someone rips a phone from your hand, throws it to the ground then and jumps on it or, say, where someone with a grudge against you smashes up your greenhouse). If the offender can't be traced, and you've got no insurance to cover the damage, you'll have to foot the bill yourself.
I'm sure a lay-by outside Sandringham or Balmoral (other palaces and castles are available in this perverse world) would be cleared in jig time with no mention of a bill.