Actually, Roman Catholicism was brought to Britain by St Augustine, and by our period England was basically pro-Rome. Plus, there wouldn't be much impetus to convert to Orthodoxy, would there? The reason all these Vikings converted was because the local rulers wanted them to, and it is always going to be more advantageous for them to have relations with France and the HRE than with a country mired in constant internecine conflict or a distant relic of the Roman Empire that's got too many of its own problems to be of any use as an ally for Scandinavia or England.
Plus, it will take a massive amount of skill and charisma to hold together three disparate kingdoms for more than a couple of generations. Even if Harthacnut lives to a ripe old age and is an ASB-Super-King, he's got three potential successors: Edward the Confessor, who was a puppet of the nobility; his nephew Aelfwine Haroldsson, who was a bloody MONK in OTL; and Sweyn Estridssen. Then Sweyn's OTL descendants fought for generations over Denmark, etc etc etc...
Essentially, we need Cnut the Great to reincarnate until such a time as two places separated by several hundred miles of chilly sea have developed some sort of national identity and a workable system of government.
The easiest answer would be to take the British Empire, add the rest of the Balearics, maybe a Mandate of Cyprus after WW1, an Anglo-Corsican Kingdom and an insanely ambitious Nelson seizing the whole of Zealand in the course of disabling the Danish Navy.