I'm an amateur historian and just love to read anything and everything I can about the Tudor era, but your question is one that I have never encountered. Poor Jane Seymour had such a short life that she sometimes gets overlooked. She came between two very controversial queens (Anne Boleyn and Anne of Cleves), and was queen for such a short time that it is easy to see why more attention has been paid to the others.
Jane Seymour's heraldic badge was a phoenix rising from a castle between two Tudor roses. Your question infers that when she chose the pheonix for her badge, Jane was somehow prophetic, that like the phoenix she would perish but rise again from the ashes. Henry VIII himself must have thought so because he had the following inscription used on her grave:Here lieth a Phoenix, by whose death Another Phoenix life gave breath: It is to be lamented much The world at once ne'er knew two such.
Perhaps that did occur in the fact that she was the only one of Henry VIII's six wives to give birth to a son who lived beyond infancy. However, Edward VI, although he did become king after his father, did not live into adulthood. Nor did he accomplish what his mother had desired: to bring the Roman Catholic Church back to England. Rather, he was a staunch Protestant and instituted the Book of Common Prayer, which is still used in the Church of England.
So I'm not sure she did represent a phoenix.