When you say a skip wagon can't reverse onto your garden, do you mean the road is fine, but he couldn't swing round and drop the skip or just that the road is so narrow he couldn't get the truck down?
If it's the former, you could dump all your waste in a pile on your garden and get a grab lorry to pick it up. They are big old beasts, but if the road is a normal size, or even single track, he can get down there, spend 10 mins picking up your waste with the grab arm that can reach over your garden wall and then be on his way.
I had my eyes opened with one of these just a few weeks ago. A builders skip is £200. You then have to barrow your rubble and soil (which weighs a ton) up a narrow scaffold plank then clean and jerk the barrow handles over your head time after time...... or you can bung 4 skip loads of soil and rubble on your front yard just tipping it on the floor and get a grab lorry for £150. Saves £650 and swearing like a sailor. No brainer.