It started early, very early. Under the Roman Empire, many Gauls, especially among the upper classes, had transformed into Gallo-Romans (who spoke a form of Latin and who viewed themselves as Romans as much as they viewed themselves as Gauls) , and it was these peoples who were first brought under Franconian control. This combined with Latin being church language in Western Christianity made Latin the language of choice among the Christian Franconian elites for many purposes, thus establishing a degree of successive cultural ties to the Roman Empire (even though Francia had no real political successive ties to the Roman Empire.)

Under Charlemagne, large parts of central Europe (including parts not formerly under Roman control) were annexed to Francia combined with the Patriarch of Rome (AKA the Pope) crowning Charlemagne Roman Emperor, further cementing the idea of Francia being the heir to Rome. This wasn't purely coincidental, the name of Rome held a lot of prestige, and in the east Rome was still a potent (albeit much diminished) force. After the Ottonian Dynasty became kings of East Francia, they (under Otto the First) got the Pope to crown them Roman Emperor, and assumed the title Caesar like the Roman Emperors (It's spelled Kaiser in modern German.)

Meanwhile, in the East, the various Russian princes had converted to the Eastern branch of Christianity (Eastern Orthodoxy) The Principality of Muscovy eventually became Russia, and proclaimed Moscow as the Third Rome, while assuming the title Czar (spelled variously depending on transliteration, like German Kaiser, it is derived from Latin Caesar.)

Now, I can't say anything for places other than France, the various countries that came from the HRE (including but not limited to Germany, Austria, Switzerland, the Netherlands and Belgium), and Russia; so I won't, but I hope this will be a good starting point.