I think it would have become nearly the same as it is today, except with a Saxon legacy instead of the Norman one.
I doubt that very much. With his conquest he was able to wipe the slate clean and restart the English Feudal system with many of its flaws removed or reduced. Flaws that France was unable to remove and which crippled the King of France compared to the power of William and his dynasty in England. That allowed him and his heirs for generations a degree of power and control that the previous kings of England (and the kings of France) could only dream of.
A native Saxon couldn't have done what William did because his supporters would have had to help him remove their own power. What feudal lord would cooperate in removing his own power and handing it to another? William didn't have that problem as his followers were gaining new lands and powers, the fact that those powers were less than English lords had previously had didn't matter to them as some power was better than none.
Without that centralization of power in the hands of the King, the country would have remained far weaker. Without that centralization there could be no Prince/King John abusing the power and the resulting Magna Carta that established the principal of a constitution limiting the power of Kings. He was able to establish the prinicpal of primogeniture, which France lacked. Without that principal the lands of aristocrats were constantly being broken down. With it there was a constant stream of younger sons to be independent knights supporting the king to gain lands and titles and the landed domains remained intact.
That lack of centralization of power was a crippling factor in the history of France. France was a far bigger, more populace country with a higher proportion of rich lands yet England was able to fight France on an equal basis, why? Because of the centralization William brought.
With centralized power Englands taxes were lower. With lower taxes peasants had more than mere subsistance levels even in the bad years. In France in bad years the peasants starved. In good years they tried to pay back the debts of the poor years. The difference allowed England to build a dispproportianate middle class compared to France. A middle class that built businesses and supported trade. A middle class that created new products and production systems. A middle class that when the "New World" was found was able to emigrate and found the colonies that built the Empire. Something they could only do because of the relatively low taxes and the higher amount of wealth left in the hands of the common folk.
I suspect that without William and the Conquest England would have been an also ran and the history of Europe and the World would be dominated by either a German or Spanish Empire rather than the British Empire and its spinoffs. The English isles would most likely have been conquered by one of the larger mainland nations, Spain, Germany or France.