I'm not convinced that asap is the way to go. It may just be my map options since I like a longer expansion phase. It comes down to which will give me the quicker early expansion, my core cities generating settlers in despotism or settler factories outside my core in monarchy/republic. The monarchy/republic debate has been pretty much beat to death a couple of times.
I'm having a hard time grasping how despotism will help you expand faster.
Elimination of the despotism penalty (+fpt, +spt) allows many additional cities to become a worker or settler pump and gives the ability to cash rush without the loss of population (aside for the other points already made). If you want a really fast expansion, leave your core producing workers and settlers too.
This is how I see the balance. You are worried primarily about your economy (based on your first post). If we are talking a pure +gpt, swapping to Republic will gain the most UNLESS you have a HUGE unit support and few luxuries to control happiness, along with cities large enough to have happiness problems (difficulty level will come into play here too). Republic will allow you to expand faster than despotism, more cities = more gpt, and that return is even greater under republic with +1gpt on gpt producing tiles.
The original question is when you can make a smooth transition between despotism to (insert other government here) and you are basing this primarily on a stable economy (my interpretation), leaving out factors of fpt & spt increases. I think that is the mistake. Though the transition may hurt financially, if you track the total advantages over the same period of time, any government will perform better overall.
There is an easy way to test this. Play a game in despotism until you find the natural transition point and then play 20-30 more turns. In a parallel game, swap as soon as you can and play to the SAME end date. Compare results. You could even see the difference by running 3 parrallel games to cover all three governments. Playing a religious civ might throw the curve a little because you will negate the early/late revolt issue but it will also remove a random element from the test. A bad revolt roll could seriously skew the results.
Connecting outer edge, corrupt cities is a distant second concern to improving the core. That is where your improvements will boost your economy and production. A lone worker roading to oblivion is okay, if your forces are 50/50 (or even close) that is a mistake (IMO) unless your core is improved to the point that no citizen is working an unimproved tile.
This is all just general, though. Specific VC make everything conditional and there are always exceptions.
