I have been to these cities two times each.
Your decision depends on what you want, of course. Culture? History? Vibe?
All three have munificent offerings in culture, history, cuisine and personality. Paris has a slight edge in culture, Rome has an advantage in history and London is a favorite in regards to energy. Energy can be a plus or minus depending on your disposition.
If it were up to me, I'd spend a bit more time in Rome. In addition to culture, history and energy, I believe Rome has the most distinctive undiscovered neighborhoods. Americans -- if you are one of us -- don't know Rome as well as they do the other two cities. Rome has a great mix of high energy and low-key neighborhoods. Another plus is that Rome has wonderful warm weather. It isn't far from the Tyrrhenian Sea.
Most people would say Paris has the best cuisine and wine, but I would vote for Rome in both of those. As mentioned, Rome has munificent offerings in culture, history, cuisine and personality.
Paris and London have all of the above, too.