Wow, Digaround, all mod cons, a shed, water butts, a sturdy perimeter fence and neighbours to deter unwelcome visitors. Lucky old you!
Suggest cover the lot in black polythene or similar, anything to block the light out, uncovering and digging it over bit by bit. If the cover is left on long enough the vegetation underneath will die off.
We're in a similar position, currently waging war on bindweed.
Good luck with your plot. Will be interested to hear how it goes.
:)
