What is the ruling on one who faces difficulty pronouncing the name "Allah" and the takbir "Allahu Akbar" due to obsessive-compulsive disorder, and does that affect their devotion in prayer?

The cure for whisperings (waswasah) is to ignore them and not pay attention to them. So, when you say the Takbeer for Ihram, suffice with a single Takbeer, and do not pay attention to the whisperings of Satan that your Takbeer is incorrect. No matter how much Satan beautifies the idea that you have left out a letter or a word, do not respond to him. The matter requires patience and striving against the self.