A busy area is pretty vague. I think downtown is the word you are looking for in American English. Here are some options in decreasing size: metropolitan area: a city and the surrounding industrial and suburban areas city: a large town (ranging from a suburban area of 100 thousand people to one like New York City with millions of people) downtown: the core or most densely packed part of a city city/town square: a relatively small area of a city that is designated to have many pedestrians walking and gathering for various activities official definition: an open (typically four-sided) area surrounded by buildings in a town, village, or city.

Downtown is specifically for the area of a city that has a lot of shops and businesses. Downtown is not necessarily busy but it can be. I can't help with city centre since I'm Ameican. I believe it's the equivalent of downtown. I do not know of a word specifically for a busy area in a city. There might be one but I'm not sure. Describing an area as busy seems fine by itself.