"There's no glacial or other explanation for the dells other than that they are 'man-made' "
Possibilities
The Lake District & The Cheviot are two areas of igneous rocks, both areas have been dramatically sculptured by the action of wind, ice and water. So I don't think you can really exclude glaciation.
Glaciation aside, running water + eons of time are enough to erode the most resistant of rock types.
It may be worth checking out the foot of the cliff where the delves occur to see if there is any evidence of running water.
Finally, the sill is an igneous intrusion that forced it's way along the strata of the host rocks. 'The delves' could possibly just be a reflection of the strata that was overlaying the sill but has now been eroded away.
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