Actually the British had done initial research in an atomic bomb in 1940 under the MAUD Committee and much of this research was given to the USA to help convince the US to develop atomic weapons. It was a joint British-American team that worked on the Manhattan Project.
The two nations had an agreement to collaborate on nuclear weapons after the war but the US was deliberately reluctant (justifiably given the circumstances) to share the information gained, mostly due to the discovery that one of the British researchers, Klaus Fuchs (a Jewish German who fled when the Nazis took power) was also a communist.
British scientists then built up their atomic weapons programme with little outside assistance to the point where they were able to test their first weapon in the 1950s.
If the British had continued their own programme instead of halting it to give their information and their researchers to the US, they likely would have had a bomb available to them around the same time as the Manhattan Project delivered its first weapon and possibly before.
For more information on how the US atomic bomb was just as much a British weapon and how the US froze out the British, refer to pages 24 to 30 of Between Heaven and Hell by Alan Rimmer
The specific pages can be read here courtesy of Google Books
https://books.google.com.au/books?id...page&q&f=false