Not to any greater a degree than in other regions of Europe. The church in England, in all its guises, has possessed national character, expressed through saints' lives, relics, and architecture. In terms of a church hierarchy the Pope was the official ruler of the English church right up until Henry VIII said he wasn't. That's not to say, however, that there weren't struggles over power and authority between church and state. The murder of Thomas Beckett, for instance (allegedly on the orders or wishes of Henry II) resulted in a struggle between Henry and Pope Alexander III which resulted in Henry II doing public penance -- a humiliating act for a king. Despite the pope's official role as head of the church, there was plenty of contention over who got to wield the real authority throughout the middle ages. This was not unique to England by any means. Henry IV (of the Holy Roman Empire) also interfered in church governance, claiming the right to invest bishops and abbots. His penance was at least as humiliating as Henry II's -- he had to walk barefoot through the snow in a hair shirt.

I'm not sure what scholar you're referring to in your question, but the separate foundation of England in the early middle ages strikes me as revisionist as well. The standard narrative of the christianization of Britain begins with Pope Gregory the Great sending missionaries to evangelize the Anglo-Saxons. Gregory's orders were to not destroy the pagan holy sites but instead to sanctify them. Now we know that there were Christians in Britain before Gregory sent his missionaries. St. Patrick, for instance, was the son and grandson of a deacon and a priest in England around the early 5th century. But Gregory's mission, along with the arrival of Irish missionaries in the 6th century, began the process of converting the kings and leaders.

If one were to argue that the English church was founded independently, I'm hard-pressed to understand where and when the missionaries would have arrived. Again, not knowing which scholars you're referring to, I'm only guessing, but perhaps they would argue that the process of christianization happened under the Romans? That's true to an extent. Christianity did trickle into Roman Britain through merchants, soldiers, etc. But saying that there were Christians in Britain is very different from saying that a Church was founded, and that may be where the confusion arose.

One more thing with regard specifically to doctrine: the Pelagian heresy, which emphasizes human free will rather than original sin, started with the British monk Pelagius around the same time as Patrick. Not an auspicious claim to fame.