WARS were fought over being the monarch of England (and bajillion other places). It's one thing to be the parent of the attempted usurper/heir named but without support, quite another to be that person. What Jane did was declare Mary the one thing she was not: illegitimate in the eyes of the 'civilized' world (Elizabeth quite a different matter); declaring Mary unfit to rule. Them's fightin' words. And Jane did not have the support of anyone other than her father, her father-in-law and, for a short time, the council. The people of England had spent years believing Mary was the eldest legitimate child of King Henry VIII, his setting his wife aside (along with Mary) was viewed as scandalous - although, since they feared their king, they did nothing - but she was the heir after Edward per most of the people, no matter what her religion (unless she 'turned Turk (Muslim)', she was next in line. Jane had no support, Edward did not have the time to promote her, the health to wed her and sire an heir (since even a daughter by the reigning kind would be ahead of Mary), or even get parliament to pass the succession. He screwed poor Jane. She never really stood a chance.
However, if you can stop her stupid father from joining Wyatt, you have a chance of (eventually) gaining her release. Then she can skip the islands for the continent without serious opposition.