Nothing would change in the 99% of medieval Arthurian romances that do not refer to any particular origin for Excalibur.
The only change I can think of is that the tale of the scabbard stolen by Morgain would not work, since Arthur did not pull a scabbard from the sword, while in the lake version, the sword held by the hand from the lake was in a scabbard. But the story of Morgain stealing the scabbard is also found in the “Post-Vulgate Merlin” which is also, in surviving texts, introduces the story that Arthur got his sword Excalibur from the lake. (Malory’s early chapters in “Le Morte d’Arthur” are mostly just adaptation and abridgement of the “Post-Vulgate Merlin” account.)
Note that the woman through whom Arthur gets the sword is only called the “Lady of the Lake” by Malory, this being one of the changes that he made to his source. But even in Malory she is not the important Lady of the Lake who is introduced later. In the “Post-Vulgate Merlin” she is simply a damsel, though one who knows magic. And she walks on the water of the lake, takes the sword and scabbard, and then walk back on the lake and gives it to Arthur. Malory also changes this, having Merlin and Arthur row out to the sword in a boat and having Arthur take the sword directly.
The damsel appears very soon afterwards in Arthur’s court, and in exchange for the sword, asks for the head of the knight Balain/Balin or of the damsel whom Balain/Balin has freed from a sword that encumbered her. Balain/Balin promptly beheads the damsel, claiming she is the damsel who had slain his father with poison. See http://members.terracom.net/~dorothea/baladro/Chapter22.html and http://www.sacred-texts.com/neu/mart/mart029.htm .
The story that one of Arthur's knights threw Arthur’s sword into a lake and a hand rose from the lake and caught it first appears in extant texts in the “Mort d'Artu”. The suggestion that this would not work without a previous story that Arthur received the sword from the Lady of the Lake ignores the fact that the damsel responsible is not known as the Lady of the Lake before Malory’s account and in any case she is long dead, and also not the source of the arm.
The story in the “Post-Vulgate Merlin” may have been known to the author of the account in the “Mort d’Artu” or it may be a later invention to provide a prequel to the account of casting away the sword. No-one knows.
The knight in the “Mort d’Artu” account is Girflet, who appears in Malory as Griflet. But in the retelling of the tales in the “English Stanzaic Morte Arthur”, the knight Beduer/Bediever and appears instead. (See http://www.lib.rochester.edu/Camelot/teams/stanzfrm.htm .) Malory knew both accounts and decided to use the name Beduer/Bedivere instead of Griflet in this case.
This is not the only account of the fate of Arthur’s sword. King Richard of England had a sword which he believed was Arthur's sword Caliburn/Excalibur which he gave to Tancred, King of Sicil,y for use on the crusades. See http://books.google.ca/books?id=irUdMNNvlakC&pg=PA175&lpg=PA175&dq=Richard+tancred+cALIBURN&source=bl&ots=NydFNSoPQr&sig=r_JUGB098GAaQTIjb-nUrMp-XCg&hl=en&ei=DScdS4GfIILd8QaL-uTUAw&sa=X&oi=book_result&ct=result&resnum=2&ved=0CAsQ6AEwAQ#v=onepage&q=Richard%20tancred%20cALIBURN&f=false . It is possible that this sword had been found buried with the supposed body of Arthur that the monks of Glastonbury unearthed.
In the earliest surviving Arthurian romance, “Culhwch and Olwen'' ( http://www.ancienttexts.org/library/celtic/ctexts/culhwch.html ) it is said:
“... and Morvran the son of Tegid (no one struck him in the battle of Camlan by reason of his ugliness; all thought he was an auxiliary devil. Hair had he upon him like the hair of a stag). And Sandde Bryd Angel (no one touched him with a spear in the battle of Camlan because of his beauty; all thought he was a ministering angel). And Kynwyl Sant (the third man that escaped from the battle of Camlan, and he was the last who parted from Arthur on Hengroen his horse).
Here Kynwyl Sant, that is Saint Cynwyl, is the last who departs from Arthur, riding away on his horse just as Girflet is the last to depart from Arthur in the “Mort d’Artu”, riding away on his horse and then forsaking knighthood to become a hermit. Note that the title “sant'' or “saint” in medieval Welsh is used freely and is applied to any supposedly holy man. There was no official canon of saints. Therefore Cynwyl, like Girflet, may well have been imagined as one of Arthur’s solders who became a hermit after Arthur’s last battle.
Whether the author of “Culhwch and Olwen'' imagined Cynwyl as casting away Arthur's sword Caledfwlch is unknown.
That the sword in the stone is an allegory referring to the discovery of how smelt iron, from it's ore runs afoul of the story of Theseus who had to learn the truth of who his father was by obtaining his father’s sword which was buried under a stone. This sword would have been bronze, not iron. The suppose allegory remains only a possibility, whether for Arthur or Björn in the Norse “Hrof Kraki's Saga”. In the Icelandic story of Sigmund, Sigmund alone can draw a magic sword from a tree.
That the Lady of the Lake may have originally been a mortal priestess is just a rationalization sometimes used by modern novelists.