Why did Paris and London develop an economic and political hegemony over the rest of France/UK whereas Berlin never grew to that size or significance?
Might not actually answer question but France and Britain have been unified countries for hundreds of years dating back to crusader times. The size and boarders of both countries has fluctuated wildly. But Germany as we know it today wasn't unified until late 1800s, after the holy Roman empire fell apart there where dozens to possibly hundreds of small city states and princedomes(sp?) That would go to war with one another often. And things like the thirty year war didn't help
Could you define what you mean be "medium city" and "megacity" or clarify your question?
I'm not sure that your premise is right. The size and population of cities are actually quite comparable between the three countries. You might be comparing metropolitan areas in France and the UK to cities in Germany?
It perhaps worth mentioning that Berlin, during the 1920's, was the third largest city in the world.
I don't imagine this answers your question, but I just wanted to challenge one of the assumptions at work.
You are probably missing the largest urban area in Germany, the Ruhr valley, which wikipedia lists at 12 million as second larges 'Polycentric metropolitan area,' between London and Paris. The Ruhr area is a area which looks essentially like one city, but it is governed as a collection of medium sized cities, not as a single entity.
On the other hand, a lot of the peculiarities of German history also seem to conspire to weaken the capital. Starting from the beginning of the HRE, where the emperor was elected by the highest nobility, which weakened his position compared to the Kings of France and England. The 30 year war, which left Germany "divided" between Catholics and protestants. And finally it is probably important that Berlin was divided into east and west, and the west actually surrounded by a wall, until 1990.
I would argue that it has more to do with national identity than anything else. In the 15th to 18th century centuries, France and Britain were the premier nation states that had strong centralized control in their capital. Up until the mid 19th century, Germany was a loose confederation of counties, duchies, and kindgoms, so it was very decenteralized. Thanks to the efforts of Prussia, Germany became unified through war and politics.