The discussion here has gotten over-complicated. For some background on the notion of "Weyl module" I should refer to my answer just posted here of an older question.
Concerning projective objects in various module categories in prime characteristic (or perhaps for quantum groups at a root of unity), it seems to be almost never true that arbitrary "Weyl modules" in such categories will be projective. Often they play instead a sort of intermediate role between simple modules and projective modules. (In the Cline-Parshall-Scott formalism of "highest weight categories", Weyl modules tend to play the role of "standard" objects.)
Note that in the category of all rational modules for a semisimple algebraic group, there are no nonzero projectives (Donkin). The idea goes back to Hochschild that in a module category for a group with additional structure (Lie or algebraic, say), the injective modules typically exist and play a more natural role in homological constructions. Jantzen's book Representations of Algebraic Groups provides a lot of evidence for this viewpoint.
I don't know enough about the spin-off concept of "Weyl module" developed since the mid-1990s by Chari and her collaborators, but here too it seems doubtful that such modules will behave like projective modules in the natural categories occurring. These Weyl modules are defined in the setting of finite dimensional modules for affine or quantum affine Lie algebras, etc.