It couldn't, assuming that the war ended in 1945 still. Not a chance, even assuming that the many and important British scientists who contributed to the US programme had been involved in a home grown one. They might have pushed it ahead of the actual 1952 date on which an independently (though incorporating knowledge from the Manhattan Project) developed British nuclear device was exploded, but not by much. I doubt that Britain could have afforded the huge investment needed whilst still fighting a conventional war.
The Soviets, also with extensive knowledge of the Manhattan Project, managed to build one by 1949 with a massive industrial effort and huge funding.
I guess the other question - If the US were not involved in a true World War, but only a regional clash with the Japanese, would they have comitted the resources to researching and building the atomic bomb?
This is a true question I have, not a rhetorical statement.