Blooms is one of the most famous Kosher restaurants in the world. Blooms is (at the moment) supervised Kosher, and not just Kosher Style. If you are concerned about Kashrut, then check for their certificate on arrival.
The original branch of Blooms was opened by the family Bloom in the East End of London after World War 2 - this was the part of town where most Jews lived at the time. It was never a "posh" or upper class restaurant, but instead served a number of traditional Jewish dishes such as Chicken Soup, Chopped Liver and Salt Beef.
More recently, as the Jewish community moved further out of London into the suburbs, a second branch was opened in Golders Green - this is an area of London with a large number of Orthodox Jewish people. The restaurant here has been refurbished, but still maintains the original idea - simple but pretty decent food, with reasonable (for a Kosher restaurant) prices.
The East End branch was still open until a few years ago, when a Kashrut scandal caused it to be closed down. This was somewhat of a shame, as although there are very few Jews left living in the East End, it was the nearest Kosher restaurant to the City of London, where many religious Jews work. Now, though, there is the Kosher restaurant at Bevis Marks.
Many famous people have dined at Blooms, and not all of them Jewish! Their website currently lists...
People marked (*) definitely aren't Jewish and have just been because of the history. People marked (?) I'm not sure. All the others are Jewish. I think the in-house menu has a much longer list though.
Don't go to Blooms expecting a really high class meal. But if you want some traditional Jewish food, which tastes good, and isn't too expensive, it's not a bad place at all. Being a strictly Kosher restaurant, it isn't open on Friday evenings and Saturday during the day due to the restrictions on cooking on Shabbat (the Sabbath). But throughout the winter it opens on Saturday night, about 90 minutes after dark.