Dallas, I have recently (2 weeks ago) been accepted in to the QRP. From the point of submission of application to being informed that we (my wife and I) had been accepted took six weeks.
The process itself took longer than this though because you need to assemble (and submit with your application) a criminal record report, medical report and proof of income (to meet the requirement of bringing in to Belize not less than US$ 24,000 per annum).
I made the application myself although there are companies (at a cost of course) that will do this for you.
I liaised with Roshel Godfey of the Belize Toursim Board and she was extremely helpful. Her email address is : firstname.lastname@example.org://www.abelizehomeforus.wordpress.com