The waiting is the hardest part! It takes months, sometimes more than a year. During this time you remain a member of the military, and you are required to follow all legal orders or you may face disciplinary action. The regulations do say that the command should accommodate your beliefs as much as possible. But, it is up to the command to reassign you and their primary concern is the military mission - not accommodating your conscience.
How others in your unit treat you while your claim is pending varies widely. Some COs receive quite a bit of respect for taking a conscientious and unpopular stand, while others experience harassment and ostricization.