What is the ruling on worshippers waiting for a sister who suffers from obsessive-compulsive disorder (OCD) in prayer until she is able to begin it? Is there a specific dhikr to be recited during the wait? Is it obligatory not to wait for her? And can this OCD be treated with the Quran, or does it require seeking help from a psychiatrist?

What you advised the sister, to abandon the whispers (waswasah) and disregard them, is correct and is the remedy. What will help her with this is supplication (dua) and turning to Allah so that He may remove the affliction from her. She should also know that religion is easy, and Allah's pleasure is attained by following His law, which means she should approach her worship while ignoring the whispers. There is no harm in seeking help from a female psychiatrist.