I suppose it depends how you define a replica. By and large the only major original piece of PS Waverley is her engine and a few sections of hull, everything else has been replaced over the years or at her major rebuild in 2000-2003.
Similarly with Medway Queen, her machinery is original but almost all of the rest of her is new save for a few bits and pieces.
Regarding rivetted ships, much of Waverleys hull is rivetted but the plates have also been welded so as to keep the regulatory bodies happy, this preserves the 'look' of the hull. I don't see why the same couldn't be done on Medway Queen, the nature of her reconstruction (perhaps a better word to use than replica) was dictated by those providing the funds, e.g. the Heritage Lottery Fund.