The population of Britain and London would be much smaller today, for a couple of reasons. Many would have been killed defending the island. Many more would have fled, first to the Free Irish State and then when that was inevitably overrun many would have attempted to flee to America or Canada. While Britain today attracts millions of immigrants from around the Commonwealth, that would not be the case if the Allies had lost the war. The Commonwealth would have disintegrated, with Japan claiming the Asian territories, Germany and Italy claiming Africa, and resources from these regions used to extend the power of the Axis nations.
Social life would have been very different. Gatherings would be not tolerated, except rallies in support of the party. Those would be frequent. Travel within the country would be limited. Communications would be monitored. And that is for those who survived the inevitable purges of intellectuals, resistors, ethnic minorities, homosexuals, artists, union leaders...
If you were Jewish, you would have fled to the U.S. Or Canada, or been part of the purges. Without Britain, there would be no Israel or it would be Palestine under fascist occupation. Either way, that would not have been an option for escape, nor would North Africa, which would also be under fascist control. Surviving as a Jew in London would require hiding your religion from everybody, including your neighbors, many of whom would be willing to turn you in, as happened in many places the Nazis took over.
If the power of these fascist states was sustainable, the resistance would eventually fizzle and we would still be living under this form of government, in a hostile world where fear was the primary means of control and average citizens had no rights. The flourishing of art, culture and economic growth that has occurred since the war may not have happened, with the focus being on powering the war machine and mass socialization to fascist beliefs.
