The machine comprises in combination: a winding cradle (7); an insertion member (29) to insert winding cores (A, A1) into said cradle; an ejector (39) to eject the logs formed from said cradle (7) causing them to roll on an unloading chute (41); a severing device (55, 53) to sever the web material after ejection of the log from said cradle. Disposed along the unloading chute is an aperture (43) elongated in a direction transverse to the direction in which the log is unloaded along said unloading chute. Moreover, the severing device (53, 55) includes a movable element (55) that is inserted into said aperture (43) to cause severing of the web material between the cradle and the finished log.