It means 'at most.' The flat has no more than three bedrooms.

"Tops" can mean "I'm estimating a value; this is the highest it could possibly be, but I think it's probably not that high." "How many vertical feet of ascent is it to the summit of Great Blue Hill?" "Oh, not that much. The elevation of the summit is about 200 meters, but the start of the trail is well above sea level. I'd say, 150 meters, tops." "How many calories in one carrot?" "Oh, almost none. Maybe 30, tops." In your example, "tops" is an afterthought, which is why it is punctuated like a second sentence. "It's like three bedrooms." [Pause for thought] "Tops." The speaker says "three bedrooms" and then suddenly has doubts that it has even three bedrooms, and adds "tops." The sentence "Oh please—the London flat is microscopic, it’s like three bedrooms," is casual, colloquial English. It's very natural, and an educated speaker might say this, but the use of the word "like" is not good English. A "translation" into strictly correct English might be: "Please don't be silly, that's impossible. The London flat is microscopic, it has at most three bedrooms; maybe fewer."