Once the tree is down, you may experience some ground "heave"
But at the very least, SEVERE pruning!
Lets hope that no one has put a preservation order on it, or you'll have reams of paperwork to complete before you dare lay a hand on it.
A good slate roof might be expensive but shoud be long lasting.
The house in which I grew up had a slate roof dating from 1923, (before my time! ) but had caused no problems when I left it in 1958, nor for a cpouple of years after when my parents moved away.
Our present house has a tiled roof, trouble free so far, for fifty years.
Rather than corrugated iron, something like Kingspan would be a good roof, and provide insulation, as a benefit.
HTH
Howard..