That is true. I doubt Henry's THAT desperate - but he really should've been - at this point.
Edward is getting one of the three: Jane Grey, Mary of Scotland or Elisabeth of France.
Elizabeth will get shipped off to either Sweden or Denmark/Norway. Unless Mary decides the marriage, then she's Duchess of Savoy.
Mary herself I'm more dubious on
Elizabeth marrying in Scandinavia would certainly be something to see, though I'm personally unsure whether Eric of Sweden or Frederick II of Denmark is the more likely match. I can see Henry VIII preferring the Danish marriage, if push came to shove, given that Denmark is older and richer, with the Oldenburgs a more entrenched family. There's also more historical precedence for the marriage, since one of Henry V's sisters married a King of Denmark.
That said, Eric was apparently
really eager for an English alliance, and the 'barely royal' Vasa family would probably be a more appropriate match for a princess of questionable legitimacy. It really depends on who is arranging the marriage, and when.
I
also thought of a fun POD for the various German matches I'd been floating around: Henry's marriage to Anne of Cleves in 1540
somehow goes well, and she's able to convince Henry that marrying his daughters off to some German princes (and maybe even some of her relatives) would be a swell idea. Heck, her
brother would actually a good age to marry Mary, and his OTL marriage(s)
technically hadn't happened yet.