William I was king by conquest, and even 70 years later ideas had changed - when Henry I died in 1135 there seems to have been no question of his eldest bastard Robert of Gloucester succeeding him (and Gloucester himself never seems to have pressed a claim!).
There is a principle of British constitutional law that Parliament can legislate on anything. this was not fully developed in 1483, but Parliament could certainly make someone a bastard. Parliament could also declare a person to be legitimate.
As far as the 'common law marriage' argument is concerned, did medieval canon law accept such a thing? Serious question, as I'm not a specialist in it. In any case, Elizabeth was born ldess than a year after the secret marriage, so it could hardly be said that the idea of Edward and Elizabeth Woodville had been living together for many years at that point.
In my view, given Edward IV's track record, it is far from impossible that there was a pre-contract with Eleanor Butler. Not certain, but certainly possible. On that basis, Elizabeth would have been illegitimate, and not legitimised by a later lawful marriage between her parents, since a second marriage ceremony did't take place. Canon law did provide that the children of an unwittingly consanguineous marriage were legitimate if neither parent appreciated that they were related within the prohibited degrees - one example is Robert II of Scots and the issue of his first marriage. I don't know whether that also applied to previous pre-contracts, but Edward would certainly have known of any plight troth he had entered into, and Eleanor Butler was still alive in 1465.
Ann