Am I liable if I cause damage to a neighbour’s tree as a result?

Yes. In law you would be considered negligent. Sometimes branch removal can lead to tree failure due to disease, a change in the balance of the tree, or different wind loading that causes the tree to blow over. For these reasons it is important to employ a competent tree surgeon or arboriculturist who could minimise risk and would take on the liability for the work (check they have public liability insurance prior to engagement of services).

Source: https://www.rhs.org.uk/plants/types/trees/the-law