In all honesty, if your Buckingham were to marry the Mortimer heir then basically the Yorkist claim to the throne becomes non-existant compared to Lancastrian and Buckinghamite(?) claims. Also the wealth and power the Mortimer inheritance gave the Yorkists would also be gone resulting in a further diminished cadet branch, if it still existed by the 1450s.
Also in real-life its pretty hard to have a constant three-sided war, eventually (in most cases) two of the sides will join together against the other.
Two of the best examples of three-way wars that lasted for multiple years would be:
1) the Wars of the Three Kingdoms from 1639-51 in England, Scotland, and Ireland with the English Civil War is the best known of them though throughout there were multiple shifting alliances between Royalist, Scottish, Parliament, and several Irish sides. Parliament came out on top after making the right alliances to eliminate other factions then later defeating their former allies differences arouse between them.
2) the Bosnian War from 1992- February 1994 when ethnic Bosniaks fought against Bosnian Croats & Croatia against Bosnian Serbs & FR Yugoslavia. Though sometimes the Croats & Serbs would work together for a few months until one side turned on the other if it was to their advantage, usually the Serbs would turn on the Croats. Then in 1994 the Bosniaks and Bosnian Croats unified their cause then along with Croatia & NATO faced the Bosnian Serbs & Yugoslavia. The war ended in 1995 with the Dayton Accords.