I guess I shall share my process:
I enrolled in college and submitted my paperwork to the college VA reps around the 9th or 10th of August. They said it would take them about 2 months to get my paperwork through as they have 300+ new veterans submitted before me. They also said that I should receive my book stipend on 1 October and my BAH on 1 November with backpay.
September 9th, I received a confirmation email from VA stating that they have received a claim from my college on September 9th. My claim will be processed in the order it was received. I hope that it means I am ahead of schedule and will receive my BAH 1 October, we shall see.