Edward VI knew that he was dying, and he needed to name an heir. One thing he didn’t want was a Catholic monarch on the throne, which would happen with his half-sister Mary. He couldn’t just jump to Elizabeth. First of all she was a woman, but second of all she was illegitimate. He needed someone who he could trust.
Since there were no males—James VI of Scotland had not been born at the time—he needed to name a female and that was Lady Jane Grey. It is likely that Henry VIII would have named Jane as queen after Edward without Katherine there to push him in the direction of Mary and Elizabeth.
Poor Jane wouldn’t have lost her head, and she may never have ended up marrying John Dudley’s son. It wouldn’t have made sense for Edward to allow something like this to happen if he knew that she would definitely be his heir. In fact, Henry may have thought about potential suitors for his niece knowing that she could possibly become queen after his son.
This would have also led to the crown remaining in English hands for longer. Elizabeth had no choice but to name her Scottish cousin, James, heir to the throne after her. The Grey line had been tainted due to Jane’s attempt at usurping it. If Henry VIII had never married Katherine, Jane would have been queen and would have possibly had children of her own. There are slim chances that it would have needed to pass onto the Scottish line for some time.