According to William Mahoney’s well researched undergraduate thesis paper from the Univeristy of Indiana, and the US Army's page on the Normandy invasion, the Allies allocated troops based on what was already available for the European theatre of operations. As far as which divisions were allocated to which landing beach, they were divided by nationality, with the British attacking Gold and Sword, Americans on Utah and Omaha, and the Canadians on Juno.

After this, the forces were allocated based on whom would be commanding each beach.

To expand: The American First Army, commanded by Omar Bradley, was charged with invading Utah and Omaha beaches. The First Army included VII and V Corps. VII Corps was tasked with taking Utah, and V corps was tasked with Omaha. The VII Corps included the 82nd and 101st Airborne divisions which played critical roles for the invasion, but were not specific to either beach.

The First Army was the army that had been allocated to the European theatre by the U.S. As such, its forces were divided among the American beaches in the American zone of control.

The British and Canadian Second Army was tasked with taking the British and Canadian zones. The XXX Corps was tasked with Gold. The British I Corps was tasked with Juno and Sword, with the 3rd Canadian Division tasked with Juno, and the 3rd Infantry and 6th Airborne Divisions tasked with taking Sword. In addition, the 79th Armored Division was assigned to the Second Army’s sector to provide armor that supported the landings.

The divisions landed jointly on the beaches, meaning that no one division was solely responsible for the "first wave". Surely some regiments went before others but it seems, from all historical records that I can find, that there wasn't a particular method for choosing which regiment went when. I would wager that each regiment was chosen to "go in" when they were ready or in an opportune position to do so.

Simply put, it was an allocation of the resources available in a manner that allowed the highest degree of control, which was the reason that the beaches were assigned to separate nations. Combining forces on invasion beaches would have added a level of complexity that was otherwise avoided.

Mahoney, W. (2014). Before the Beaches: The Logistics of Operation Overlord and D-Day. Retrieved July 30, 2017, from https://spea.indiana.edu/doc/undergraduate/ugrd_thesis2014_mgmt_mahoney.pdf

http://www.history.army.mil/brochures/normandy/nor-pam.htm

EDIT: Added the source that I forgot to add. Also added a bit to properly answer the question.