Considering Felipe II of Spain's popularity in England, it's not impossible for the english parliament to bypass Mary's will and choose a protestant regent for Henry IX againt his father, no matter how heartless it is to separate a father from his son. Besides, the Scots did that later in OTL with Mary, Queen of scots and her son James I/VI (although it was more the nobility than the parliament).
However, we're talking about Felipe II : I doubt he's going to give up his son, especially to a protestant regent.
Elizabeth's role will probably be important in the next things to come as she is the closest to the throne after her nephew...
Even with an ATL Philip II, this would be more unlikely, than the OTL scottish situation with James VI/I.. Besides Philip II is supported by 'Spain' etc. and catholics and (at least some) royalists, the OTL situation in Scotland was totally different.
The English situation ITTL doesn't put a certain side in a decisively dominant situation.
Although even a peaceful regency council would have to be approved by parliament, but Philip II could have a say in this process.