In a winding assembly for winding web rolls, an enveloper assembly is used to initiate winding of a web stream onto a pre-formed core or to initiate winding of a core-forming substrate into an in-line core. The enveloper assembly comprises a first support arm operatively coupled to a second support arm; a third support arm operatively coupled to the second support arm; and an enveloper roller operatively coupled to the third support arm. The enveloper roller is movable along a cylindrical surface from a first position to a second position. The enveloper assembly accommodates a wide range of core diameters. For winding non-adhesive web streams, a tail tucker can be used in conjunction with the enveloper assembly to initiate winding. In conjunction with apparatus for inserting core-forming substrates or adhesive tabs onto a web, the winding assembly can perform high-volume, streaming production of wound web rolls.