A web turn-up cutting method and apparatus for severing a continuous web has a water jet controller adapted to activate a water supply of first and second water nozzles and a carriage controller adapted to actuate the nozzle carriages to provide a transversal movement of the first and second water nozzles. The first and second nozzles are adapted to define together a substantially X-shaped cut on the traveling continuous web defining at least a substantially V-shaped tail for ending a forming roll, an opposite substantially V-shaped start for starting a new spool, a detachable first wing and a detachable second wing.