Even before the fall of the Soviet Union, Ukraine and Belarus (which were then officially part of the Soviet Union) had independent seats at the United Nations. Why didn't the US or the UK get the same privilege?
The Soviet Union had three seats in the UN. In addition to the Soviet Union itself, two of the Soviet republics had seats: Ukraine and Belarus. This obviously didn't make much sense given that neither of them was an independent state at the time. So it can only be viewed as a way for the Soviet Union to increase its weight in the UN.
This was one of the results of the Yalta Conference in 1945. In fact, originally the Soviet Union wanted that all the then 16 republics are represented, in the end they settled on two. Why did Roosevelt agree to these concessions? I can see the following reasons:
Wikipedia claims that the USA also had the option to get two US states represented in the UN but this didn't happen for political reasons (difficult to choose). It's a very weak article not listing any sources so one would need to find better evidence.
As to the UK, in a way they got even more than three seats with Canada, Australia and India present. Note that India wasn't even an independent country at that point.