Federations in McQuaid's native country, Ireland, and Switzerland, where he lives, withdrew their support.
Cookson's legal advisers believe UCI statutes require a presidential candidate to be nominated by his home federation.
American officials had support from Russia, Canada, Finland, and Algeria to demand clarity from the UCI via a CAS ruling.
The UCI said McQuaid, one of four executive board members, took no part in the decision announced Friday. The board includes UCI vice presidents Cho Hee Wook of South Korea, Renato Di Rocco of Italy and Artur Lopes of Portugal.
McQuaid is seeking re-election amid a reputational crisis for cycling's world body in the fallout from the Lance Armstrong doping affair.
Cookson, the long-standing president of British Cycling and a UCI management board member, pledges to restore the sport's credibility.
The winner is chosen by secret ballot of an electoral college of 42 delegates.