When people refer to the fall of Rome, they ware usually referring to 476 AD, when the last West Roman Emperor was deposed, and the East Roman Emperor was asked to unite the offices of the two emperors. The fall of Rome was a long, drawn out affair that actually began well before 476 and lasted until well after. And my guess is that at the time, very few people noticed any sudden change between how things were before and after.
The government of the Roman Empire continued to survive in Constantinople until 1453.
The Senate of the West Roman Empire continued to function separately from the East Roman Empire until 603. It may have continued after that, but that is the last date for any action we know of it did.
Clearly the Church continued to function. And so did the governments of all the various Germanic kingdoms that had already sprung up: Franks, Visigoths, Ostrogoths, Vandals, and so on, for varying amounts of time.
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