I was actually thinking of Henry marrying Mary. Either one really, but the first Henry would be closer in age so he might be a better choice for a husband. Henry marries her of to her cousin in a attempt to keep her claim to the throne in England, and who better for her to marry then the son of his best friend? Or during Edward's reign, Mary and Henry are married as an attempt to counter the Protestant reformation in England, perhaps spurred on by the Emperor. As for Lincoln, I didn't even think of a Boleyn match, considering how much his mother, Mary Tudor, hated the Boleyns. Though if Henry fails to have children (with either Mary or a different wife) then would Lincoln ( or his descendents, hopefully a son) inherit the throne when Elizabeth dies? I mean legally the Lincolns (for lack of a better term) would be the heirs by Henry VIII's will but Elizabeth's advisers engineered James VI's ascension to the throne OTL would that still happen OTL or would Lincolns heirs become King/Queen?
As a legitimate English-born nephew of the King, Henry Brandon would make a better candidate for Mary's hand than James of Scots. King James was briefly considered OTL but Henry was more interested in marrying Mary to the Emperor, with the end result that James turned to France for his two queens. I don't see either Mary or Lincoln ever going abroad during Henry VIII or Edward's reign - Lincoln especially would have more to loose than gain. More likely than not he would tread the line and not risk his head opposing his uncle's break with Rome or the Protestant reforms of his young cousin.
If (as seems likely) Henry still prevaricated over Mary's marriage, young Henry would probably marry Catherine Willoughby, a rich heiress whose wardship his father obtained (for him) in 1528. Another rich bride would be Anne Bourchier, Countess of Essex, although she married William Parr in 1527 so I'm not sure what would be necessary to break that off. Henry VIII's will would have explicitly designated him heir following Mary and Elizabeth. By the time Edward came to the throne, he'd most likely have a son or two - say, for example, a son born in 1536. This son would most likely be married to Mary or Elizabeth in the twilight of Edward's reign, neutralizing that threat; Elizabeth would be the more attractive bride, although the fear of foreign intervention on Mary's behalf might make her a better choice, especially if the Brandons harbour Catholic sympathies.
If Henry or his son marries Mary, then I'd say Elizabeth is excluded for good (similar to Princess Anne and William III OTL), as he'd no doubt posit for the crown matrimonial. Assuming Mary died more or less at the same point as OTL, her husband would almost certainly marry the widow Queen of Scots c.1561, excluding Elizabeth definitely from the succession.
EDIT: Tl;dr: A surviving Lincoln would likely become King long before 1603.