I believe that simply chopping it down could cause more harm than good, ie
if there are roots under the property which then rot away leaving
voids...
can anyone advise what I should do? Chop it down in stages? If so, by
how
much? and over what period?

You should be OK. Subsidence is often caused by the tree sucking large
amounts of water out of clay soil. The a tree of the size and species you
have should not be a problem. Cutting it off level with the ground is best
since you will not disturb the roots. They will take ages to rot so don't
worry about them. It would be a different matter if you had something like a
willow that is the thirstiest of the lot.