Will a person suffering from obsessive-compulsive disorder be held accountable if they delay the prayer until its time has expired due to their obsessive thoughts, and what is the recognized time for the Fajr prayer to expire, and is re-performing ablution obligatory if the person suffering from obsessive thoughts feels that the water did not enter the nose in sufficient quantity?

You must strive against obsessive thoughts and get rid of them. Sunrise means the appearance of its first part, not its complete rising. What happened, being preoccupied with ablution until sunrise, we hope that Allah will forgive it, as it was under the influence of obsessive thoughts. It was possible to perform tayammum (dry ablution) according to some scholars if one feared that the sun would rise before completing purification. Rinsing the nose (istinchāq) is a Sunnah, not an obligation, in ablution according to the majority of scholars, and it is achieved by water reaching the inner part of the nose.