Originally the Common Law regarded the Realm as just another (very large) fief, and it followed the German law (not Salian Franks). Which is called male preference agnatic primogeniture, I think.
Descends to the oldest son of the most recent holder, then next oldest son etc , in the whole blood , then eldest son of the previous holder, and working back so long as there is a male descendant of the first holder. If that fails, repeat for the half blood. Then if there are no male descendants at all of the first holder, repeat process with daughter.
That later got changed for ordinary fiefs to have multiple female heirs hold as co-parceners. That change was never picked up for the realm.
The Salic law never applied in England.
English succession law (like most succession laws at that point really; disputes tended to be resolved on the battle field, see e.g. the Hundred Years War) was never really codified until much later. If you go through the English monarchs, you have many cases of the person who would be the logical successor under male-preference primogeniture being ignored; the Anarchy is the most famous one, but the origin of the Yorkist claim involves another one.
Regardless, one of the Lancastrian justifications (beyond the obvious one that Henry IV had the power to make his claim stick back in the day) was that Edward III had allegedly attempted to change the succession law. Since the House of York was descended from Edward III's third son through the female line, they were senior to the Lancastrians (descended from the fourth son through the male line) according to modern inheritance rules; hence the confusion. (Edward's second son died in infancy, and his first son's line had died out, with some assistance from Henry IV).
In practice, as the Wars of the Roses went on, that changed; Henry Tudor, the successful Lancastrian claimant, was famously descended from a legitimized bastard of the fourth son through the female line (and his mother, who by modern practice would have been the queen, was still alive), but by that point there were no real male-line descendants of the House of Lancaster to provide an alternative.
To get back to the OP's question, I agree with what was said earlier: the new succession law would continue until some future Lancastrian king was left with only daughters. Whether the result would be more similar to the OTL accession of Mary I or the the Empress Mathilda will depend on the circumstances.