Depends on how strictly you're going to follow RAW.
As written, shillelagh applies to clubs and quarterstaves, not greatclubs. If you convince your DM that greatclubs should be included in the spell's legal effects, then you've got to convince them that a greatclub is heavy since it lacks that qualifier (though it really should have it since it's quite a bit heavier than several other weapons designated as heavy).
Though, really, the first provision of GWM applies to any melee weapon, so a club or quarterstaff enhanced by shillelagh would be affected by it in any event as far as I can tell.