I think this may be an urban myth. Henry sent out emissaries (well stocked with gold in their purses) to "visit" various scholars and clergymen throughout Europe whom he hoped might help him put together a theological argument that would support his call for an annulment of his marriage to Catherine of Aragon, thereby forcing the Pope to give in and grant him what he wanted. I think some biographers may have a stretched this a little to include the Orthodox Patriarchs but there's no evidence to suggest that Henry ever sought their advice. Cromwell did present the example of the Byzantine Emperors to Henry as an example of how the new relationship between Crown and Church might work but I don't think this inspired Henry in any way to look seriously at conversion to Orthodoxy.
Henry's argument at the time wasn't that the Pope was being an old meanie who wouldn't let him marry that pretty Boleyn girl. The whole thing might have been inspired by Henry's demand for an annulment of his first marriage but it really became an argument over who actually called the shots in England - the King or the Pope? And as part of that argument, Henry relied on a 14th century law known as the Statute of Praemunire which made it a criminal offence for legal cases which could be heard in England to be heard in other jurisdictions. In other words, the Pope had no authority to rule on matters of English law.
How Henry interpreted this is important to your WI because he used it as justification to penalise the English clergy for taking orders from Rome. And that then led to him install himself as Supreme Head of the Church of England. Whilst Henry wasn't averse to a bit of hypocrisy here and there, I highly doubt that he'd take the actions he did based on that premise only to have the English clergy defer to a new religious authority figure abroad. Of course, he could have established an autocephalous Orthodox Church in England with himself as it's head but I think that too was unlikely.
The Church of England in Henry's day wasn't the Church of England today - though you can easily find traces of it in so-called "High Churches" where you'd be forgiven for thinking you'd wandered into a Roman Catholic Church instead. The theological differences aside, Henry made the Church of England an easy switch for his subjects by keeping so much of the role of the church in their daily lives the same so as to make the transition smoother and more readily accepted. Orthodox worship was so different to what anybody in England was used to that I think it would have proven too chaotic and too alien for many to get on board with.
A very interesting WI though!