There are probably a few different reasons why the U.S. didn’t include many other supporting countries in the Manhattan Project. One reason is that the United States probably didn’t have full trust in the USSR. They had originally sided with the axis powers. Those countries included Japan and Germany and they actually helped take over Poland before 1941. In 1941, they moved to the allied powers and joined with the United Kingdom, Canada and the U.S.
There were several other countries that were part of the allied powers and had expressed loyalty to that side. The USSR was not the only country fighting with the allied powers that did not know about the Manhattan Project. It was kept a secret from other countries as well.