Kristina of Denmark famously said she'd marry Henry- if she had 2 heads.
Two new questions:
1) Besides Henry being Henry, and Wolsey once remarking that "one should tell the king what he SHOULD do, not what he CAN do", why didn't he go for the public honesty option? I mean if it meant that Karl V is more likely to side with you than his aunt, then surely it was (at least in hindsight), the better option
2) As to the foreign candidates, following Anne's sword accident and Jane's death, France offered Marie de Bourbon, Marie de Guise, (I think I read somewhere Marguerite de Valois was also proposed as part of a double match), and one or two Lorrainer/Bourbon princesses. In response, the Imperial camp offered Kristina of Denmark or Maria of Viseu (however, both imperial matches foundered on the fact that both ladies offered were great-nieces of CoA, so the affinity would've required a papal dispensation). And the other option was that both France and Spain requested a double match, Henry to the lady, while Mary to either the Infante Luiz or a French prince (dunno who the French wanted after the duc de Bretagne died), PLUS her reinstatement in the succession as legitimate.
To answer the first question:
Well Wolsey was keen but of course it effectively meant that Henry would have had to go back on his word that Catherine had not been a virgin on their wedding day - which would have made him look very bad - effectively he would have been admitting he had lied.
I suspect to Wolsey it was probably a more honest course that allowed a degree of respect towards the Queen (accepting her arguments about her virginity) and was certainly more likely to be view sympathetically across Europe - he was probably grasping at straws knowing that the political issues alongside Catherine's determination to remain Henry's wife would mean any case was likely to fail.
Secondly it is highly unlikely that the Pope would not have been aware of the existence of this impediment given Catherine herself was claiming to have never consumated her first marriage.
Although most people traditionally viewed that affinity only followed a sexual relationship some in the 16th C did consider cohabitation as man and wife enough to create affinity irrespective of whether the marriage had been consummated or not which might have been what Wolsey was hoping for.
However it would still have been weak given the original dispensation for affinity used the term "perhaps" with regard consummation so the Pope could still argue that there had been no need for a second dispensation of public honesty and the marriage was therefore valid.
To be honest Henry was doomed due to the political situation - and ugly annulment cases were not unusual look at Louis XII's (and his case was just as weak as Henry's) annulment from Joan of France that was appallingly embarrassing for all parties involved. Henry simply didn't have the power, the prestige or the influence in Rome to convince the Pope to override a previous dispensation.
Without Anne you could argue that Henry would have remained a loyal husband to Catherine - and plenty of people at the time believed that Anne was the root cause (or rather Henry's fascination with her) of his appeal to Rome for an annulment - however given he was still without a male heir and Catherine was still living - there is always going to be chance he would move for a new wife at some point.
Edward VI* Born To Anne Boleyn
Continuing with this thread, and Anne buying herself time by being delivered of a live male son in 1536, instead of miscarrying in January. The baby is christened Edward (Henry seems to me, superstitious, so he wouldn’t name the child Henry, since all of his sons named such had died (Fitzroy according to Chapuys soon to be the latest of them) with much rejoicing. Anne, however, dies in May (her OTL death date, simply with the CoD being childbed fever rather than an accident involving the headsman from Calais).
Henry VIII, being Henry, then marries Jane Seymour (maybe a bit later in 1536 or in 1537), who either gives him another child, before dying in childbirth. The rest of Henry’s marriages proceed as OTL (Anne of Cleves, Kitty Howard, Kathryn Parr (maybe he can still marry the Dowager Duchess of Suffolk as he intended to do if he managed to put Kathryn Parr aside), and he finally dies. Edward VI then succeeds as king of England. He rules until 1553, when he dies (pick a way, how – although Elizabeth was seemingly the only healthy one amongst Henry’s children (Mary and Edward suffered from congenital syphilis, while Fitzroy was tubercular) ) without a child – mostly because there’s a similar shuffling of feet about who he can marry (chief candidates: Élisabeth de Valois, Juana of Spain and Mary, Queen of Scots).
Now, regardless of the butterfly herding that this would require, how does this young Edward grow up? Henry flipflopped between Catholic and Protestant policy (while AFAIK he saw himself merely as a Catholic with a few papal issues) for most of his reign (particularly shown by his choice of wives: CoA, Jane Seymour and Kitty Howard were all Catholic (or Catholic favouring), while both Annes, Katheryn Parr were either Protestant or Protestant-leaning; Katherine Brandon née Willoughby IDK, since her mother, Maria de Salinas, was a Spaniard from the retinue of CoA, whilst she was one of many Marian exiles during and a friend of Kathryn Parr), so chances are that Edward grows up in a similar environment, religiously speaking, to his OTL namesake.
However, what does this future hold for Mary Tudor, the Lady Princess Elizabeth (mom doesn’t get executed for treason, parents don’t divorce, can’t see why she would lose the title), and Jane Seymour’s sprig (if a girl, or the effects on tempering the Boleyns if she has a boy (who could be just as sickly as OTL Edward VI)?)