If the Council are like most councils they will try everything to avoid removing the fly tipping if it is on private land.
Looking at the material in jimit's picture it looks like bricks. The best way to remove these would be by JCB (but that would then need a wagon) or by using a crane mounted on a wagon. Either method may damage the site.
Of course if the site is not scheduled then the only legal recourse I would guess would be criminal damage but the Council would only remove the fly tipping with the permission of the landowner. The Council then have a statutory defence to criminal damage.
Grimly,
G