Didn't Frances Brandon die in 1559, prior to the POD? If she is still alive and in the running, there might be some unease regarding her rather lowly second husband Adrian Stokes.
If Frances is dead, the big consideration is the youth of so many key figures- Mary, Francis, Darnley, and Katherine Grey (who hasn't yet discredited herself with the secret marriage to Seymour) are basically all teenagers, so much of the power will be in the hands of the adults around them. The only full-fledged adult with a stake in the succession is Margaret Douglas, who will undoubtedly be politicking on behalf of herself and her son.
Much would depend on the nature of Elizabeth's demise- how quickly does it happen? How quickly does news of it spread? Who do the people at the heart of her regime favour?
I'm with Kellan on the English being rather leery of being ruled by a French king, which makes things difficult for Mary. How much Mary and Francis could actually do to press their claim to England given their own domestic troubles may also be in question.
The best case scenario is probably fear of the French pushing the various figures in England into some form of cooperation (perhaps with a Darnley-Katherine marriage) and a moderate, conciliatory, road in terms of religion- going full fledged Catholic would undercut most of Mary's potential support, but that might stir up Protestant unrest, especially given the Protestant members of Elizabeth's regime that wouldn't like being turfed out of power. In any case, putting Darnley anywhere near the throne could end in disaster- Darnley is Darnley, after all. And Katherine didn't display the best judgement IOTL either.
Elsewise, you might get the Protestants making a power play with a claimant of their choice (Katherine?) and trying to exist on a rather slender base of support against the inevitable Catholic pushback. But Catholic loyalties might be divided as to who to back, the French will be stirring up trouble in Mary's name, but Margaret Douglas will also be intriguing, and there's the Spanish floating around in the background who will be pro-Catholic but anti-French.