This idea was inspired by the East Orthodox basically claiming Harold Godwinson as one of their own.
It's essentially a fringe thing from Orthodox communities in England, whom a good part doesn't even suspect it's a thing, I gather.
It's approximately as credible as a 3€ banknote.
The whole narrative is built around William being depicted on a semi-Crusade of sort against Harold and the church of England which have in the best of cases very little ground to get based on. While IOTL, Norman clerics were quite active in Rome to justify and get William's claims legitimized (and really went to every court they could), it's not really sure how much the pope supported William's expedition besides accepting the legitimity of his royal claims, while Harold's passivity on this matter and relatively bad reputation didn't help there wasn't a real problem between Rome and the English church that didn't existed elsewhere : the pontifical support is real, but it's not even a given the banner sent to William was representing a sanctification of the war (as it appears in Bayeux Tapestry, it doesn't looks St. Peter's banner) rather than a papal dispensiation to undergo a war as it was justified along a "Truce of God" perspective (and was used by other rulers during Alexander's II reign)
http://santlache.blogspot.com/2010/09/pope-in-1066-william-or-harold.html
What if Harold or one of the other kings sent emissaries to Constantinople to switch religious affiliation from Rome? How will if influence England? Would there be a noticeable Greek influence in not just religion, but also language, culture, and so forth?
In spite of more or less former disputes and growing estrangement since the VIth century, you didn't have something very clear as Catholic vs. Orthodoxy and rather something like Latin vs. Greek Christianity. If the necessity arised to cut ties with Rome, you'd probably end up with an excommunicated ruler eventually dealing its own business as it happened in the Xth and XIth centuries, and if William would have failed to conquer the kingdom, you'd have such situation with eventually Harold making penance and being admitted back to the ensemble of Christian kings.
The main difference would probably be a delayed introduction of monastic reforms and stronger episcopal power.