I'd agree that Henry ends up Lord Protector. Katherine might make a play for the Regency, but I don't see the English accepting it. Elizabeth would be the first Queen Regnant since Matilda and the anarchy still rang clear in the people's minds; the War of the Roses are remembered too: Elizabeth of York was the clear successor to Richard III, yet Henry VII was the one who became King. He still married her to legitimize his reign, but it shows even in this period the English were distrustful of female succession. I think little Elizabeth has a chance though with Katherine making enough noise, and she and Henry were quite close, so if he supports her, that would seal the dea--and I think his support would be contingent on having a say on whom the little queen marries, with intention clear she will marry one of his young sons.
I can see Katherine being appeased with a place on the council; I suppose she could also ceremoniously be named Regent with the Duke of York exercising power as Lord Protector, but the fact is, at this point in time, the Lord Protector often was the Regent. The first real "Regency" wasn't until Hannoverian times, when George IV served as Regent for incapacitated father. I suppose it could be arranged as such, but I find that doubtful. Henry as Lord Protector and Katherine with a ceremonial seat seems more likely.
As for our Lord Protector, he's going to want to contract a match as quickly as possible. A French alliance could be possible through Navarre -- Queen Catherine had a multitude of children. I unfortunately can't think of many other princesses off the top of my head --possibly Françoise of Alençon upon the death of her first husband? But given the need to marry quickly, he might even seek out a match amongst the German principalities. Unfortnately it's quite hard finding a princess close to his age: his future son is already going to be some years younger than Katherine's daughter.