If the branches hang over your property, the roots almost certainly will be under your property. Conker trees can grow huge, so unless this is a country garden with acres of space to spare, it seems somewhat unreasonable to have such a huge tree in a garden at all. You had better check there is nothing close to the tree on your side which might be damaged by roots. The law actually states, unless it has very recently been changed, that you may only cut the neighbour's overhanging vegetation if they have unreasonably refused to do it themselves, when asked. But it may well be protected, as perhaps it should be. Local council will tell you if there is a protection order.
Near me there is a house which used to have conker trees; the owners tried to prune, but ended up having to fell, because the pruning was unsuccessful. You might well consult the Citizens Advice Bureau, or you may end up with more than you bargained for.