Q. Does the annealing process start as soon as the heat starts to drop and is it relative to the thickness of the glass?

You can find many versions of the annealing process.  

Charles Bray, Ceramics and Glass: A Basic Technology 
Henry Halem's Glass Notes: A Reference for the Glass Artist (the 4th edition relies on Woolley's writings) 
Frank Woolley's Glass Technology for the Studio 
Keith Cummings, Techniques of Kiln Formed Glass 
Frank Woolley's chapter in Lucartha Kohler's Glass: An Artist's Medium 

Woolley was a research scientist at Corning Inc. For several years, he took glassblowing classes and spoke to The Studio students about annealing and other elements of glass technology. Many glass artists have adopted his suggestions. 

Briefly, from Halem, p. 40: 

"The object of annealing is to remove these gradients by soaking at a constant temperature within the annealing range for a suitable length of time. The time relates directly to the glass properties and thickness. It should be noted that one can anneal for the proper length of time for a specific form but strain can be reintroduced during the cool down from the annealing point to the strain point [the transition range] if the work is not subjected to equal cooling rates at any given time. The cooling rate is dependent on glass properties and thickness...." 
"The optimal annealing temperature is usually, but not always, 50 degrees F below the softening point of your glass. Glass cannot be annealed below the strain point. Although glass can theoretically anneal at any temperature in the transition range, it is most efficient to anneal at the high end annealing point." 

As "Dr Glass," Henry goes on to explain reasons for not annealing glass properly: 
- "...the total piece of glass has to be brought to thermal equilibrium at the annealing temperature before you begin to cool it. The thicker the piece of glass, the longer it will take. 
- If you anneal far enough below the annealing point you will not relieve the glass of enough strain in the time allotted. 
- [If you cool] your glass too quickly through the transition stage [you will] reintroduce strain." 
He also mentions incorrect pyrometer readings, temperature gradients, and other factors. (p. 41) 

On p. 42 of Halem's book, Woolley states that annealing "...is accomplished by first holding the piece in a temperature range where internal stresses are relieved (without being so hot that the piece deforms under its own weight), then by cooling slowly enough that the permanent stresses left in the piece are acceptable for the application. 
...it is imperative to design the [annealing] schedule for the glass composition and shape, as well as for the acceptable level of permanent stress in the piece." 

"A Typical Annealing Schedule Has Four Parts: 
1. Fast reheat to annealing temperature (only needed if pieces have been cooled without adequate annealing after forming). 
2. A soak slightly above the annealing point to relieve forming stresses. 
3. Slow cooling to below the strain point. 
4. Faster cooling to room temperature. 

- The most time-consuming part of the schedules the annealing and slow cool through the transition range. 
- The times and temperatures in the annealing schedule depend on the thickness of the piece and on LEC of the glass. 
- The rate of cooling is inversely proportional to thickness squared so thicker ware must be cooled at a slower rate. 
- The rate of cooling is directly proportional to the LEC, so high-expansion glasses have to be cooled more slowly."