First thing's first, I am aware that holding both positions simultaneously would be a huge conflict of interest and would almost certainly never happen. My question is not regarding whether or not that would even be possible.
I had two different professors assert both of these about Reginald Pole:
Both of the professors basically said that Reginald Pole was next in line to become the King of England and Pope, but was prevented (or passed over) for each position. My question seeks to address the veracity of each claim. If I can clarify anything, please let me know. Also, if I can organize this better let me know how.
While I don't know about the Papacy, Pole was definitely nowhere close to the crown of England. His best hereditary claim to the crown came from his being the grandson (in the eldest surviving line) of George of Clarence, younger brother of King Edward IV of the House of York. However, the House of York had been defeated at Bosworth Field by Henry VII of the House of Tudor. One could still theoretically advance Yorkist claims without being called traitor by pointing to Henry VII's marriage to Elizabeth of York daughter of Edward IV; Henry VIII felt threatened enough by these claims to have Reginald Pole's older brother beheaded. Still, that was more a matter for war than for law.
So, no, the crown would not have legally gone to Reginald Pole. By Act of Parliament and Henry VIII's will, Edward IV's heir (as he died without issue) would be his half-sister Mary. Edward himself attempted to change this, ruling her a heretic and bastard (as she was, from an Anglican standpoint). The legal heir then would have been Frances Grey granddaughter of Henry VII, or if she had been excluded on the grounds of her parents' questionable marriage, Mary Queen of Scots. In fact, Edward selected Frances Grey's daughter Jane, who was after nine days' reign arrested and later executed by Mary. But all this goes to say that there were many other heirs - Pole would not have been legally anywhere close to the top of the line.