I'm on an estate where there are lots of trees, and any bigger than a certain size automatically have a preservation order on. In our old house we had a beautiful beech tree, taller than the house, with a trunk about 5' in diameter in our 15' garden.
There are a beautiful set of very tall trees (over 30m tall I'm told) through the estate, and houses are built around them.
Some insurance companies will not insure any house on the estate, I'm told. Even our current one, which had no trees that will even tickle the outside edge of the house if they fall directly towards it.
However, we've never had a problem, and no one has not been able to get insurance at all.
Trees do sometimes come down. We had one storm where 4-5 came down, two landing on houses/garages, but it's not common. It's not always predictable, but if you keep them checked regularly, you'll bring the likelihood down.
The biggest issue is that for some houses the roots get into the drains, so it will be worth getting your surveyor to send cameras down. Again, it isn't a problem we've had, but it's the biggest one round here.