which organizations were involved in it?
E-9 group,  National Safety Council, etc.
Trinity (nuclear test)
Bainbridge's group was known as the E-9 (Explosives Development) Group. Stanley Kershaw, formerly from the National Safety Council, was made responsible for safety. 
what is trinity nuclear test? It was the code name of the first detonation of a nuclear device. when did the test start? July 16, 1945. did it have any other code name? No
wiki:5358139
Trinity (nuclear test) [SEP] Organization
In March 1944, planning for the test was assigned to Kenneth Bainbridge, a professor of physics at Harvard University, working under explosives expert George Kistiakowsky. Bainbridge's group was known as the E-9 (Explosives Development) Group. Stanley Kershaw, formerly from the National Safety Council, was made responsible for safety. Captain Samuel P. Davalos, the assistant post engineer at Los Alamos, was placed in charge of construction. First Lieutenant Harold C. Bush became commander of the Base Camp at Trinity. Scientists William Penney, Victor Weisskopf and Philip Moon were consultants. Eventually seven subgroups were formed: The E-9 group was renamed the X-2 (Development, Engineering and Tests) Group in the August 1944 reorganization.