I'm having difficulty understanding your question, could you please expand on it? Do you mean when the Gothic Kingdoms stopped paying lip-service to Constantinople, or the revival of the title of Emperor in the West with Charlemagne, or even after 1453?
There’s the early 6th century period where the Ostrogoths stopped pledging to the Empire, and among the reasons why Justinian I waged the reconquest. But the establishment of the Exarchs soon after in Italy and North Africa reversed this until the domination of the Franks in the 8th century and the Islamic conquest of North Africa in the late 7th century. Basically, a complete reversal of Justinian’s gains aside a few losses like Egypt and the Holy Land.
However the Iconoclast civil wars , the loss of territory and the resulting rise of Irene of Athens , after blinding her own son, all weakened Constantinople’s credibility as the leader of Christendom. Why? The loss of land had a religious bent that it was due to sins and moral failings. However we cannot forget the growing discontent that people outside Constantinople had towards the Emperor due to their mismanagement and political blunders that made it unrealistic if not unmanageable to consider them an authority at all. And Spania, a remnant of Hispania, fell to the Visigoths in 624.
The Italians looked to the Ostrogoths and then Franks as their authority, crowning Charlemagne as the ultimate middle finger to an Empire that cared more about petty bullshit like fighting over mere finger paintings* mixed with the Salem Witch Trials than protecting its extent territories and stability that they abused and neglected.
This implies that most of Western Europe was no longer recognizing Constantinople as its leader by 800. Britain or whatever was there obviously sided with the Franks, others under Saracen or Moorish control and so on.
Then the Empire still kept Venice, Southern Italy and had Sicily until the 12th century with the ascension of the Normans while Venice broke away as its own authority after the Fourth Crusade in 1204.
*Thats my jab at the nonsense.
These are all the times people broke away from the Roman Empire.