The determination for who will be among the 1800 exempted Torah scholars will be managed by a committee of the Roshei Yeshivos. If the Roshei Yeshivos are unable to work with each other on this and agree on a decision, then the responsibility will fall to the Minister of Defense.
Whether one qualifies as “Chareidi” for purposes of the quota is primarily determined by whether one has attended a Chareidi yeshiva for at least a certain amount of time. The law includes criteria for what counts as a Chareidi yeshiva.