the only impediment was that there was no proof that Catherine Grey's marriage was legal - she lost "a piece of paper", the witness included her sister in law who was dead by the time the marriage emerged, etc - her real offence was that Elizabeth didn't like her very much - and that Seymour's offence was marrying someone close to the throne without consent. Fairly easy to blame it all on Elizabeth being vindictive and finding a witness (the Archbishop says oops we made a mistake) to say it was valid - James VI and I had no problem accepting the marriage was valid once he was safely on the English throne.
As i said it depends entirely on whether the council want a) a child monarch b) a child monarch in their control or c) the protestant unification of England and Scotland.
Seymour wasn't particularly unpopular at court - this is before his second dodgy marriage - though he would be easy enough to control and it is unlikely he would be made sole regent for his son.
So Elizabeth dies February 1570 after a short illness - the Queen aware of her impending death - names her beloved Dudley "protector of the realm" - when asked to name an heir though mumbles about he "who has the most right" - a cabal of the council approach the Earl of Hertford and take custody of his son - with the intention of proclaiming him King as Edward VII - others favour the Scots King.
And the great dance begins.