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.
The "original" UN consisted of countries in several distinct categories: 1) the "Big Four" of America, Britain, the Soviet Union, and China 2) British "Commonwealth" countries such as Australia, New Zealand, Canada, South Africa and others 3) Latin "American" countries in Central and South America and 4) "captive" nations under Nazi rule with governments in exiles like Poland, Greece and Belgium.
Britain had multiple nations in the UN if you count the Commonwealth nations, and likewise for America if you count nations from central and South "America." So in this regard, it seemed only fair that the Soviet Union have several nations in the UN, with Russia, Ukraine and Belarus being the three largest ones.