That's a very good question.
My experience has gone short and long. Two months ago I added a piece of dry, dead, branch LR into a 5 year old system. The rock had been sitting outside for a year in the rain and snow. I hadn't paid much attention to it but just yesterday I happened to notice pink Coralline Algae was already starting to cover it. That means it took less than a month for it to start being populated by algae, bugs and worms. Of course this rate of growth is unusual, probably due to the maturity of the tank and the fact that Coralline Algae completely covers the back and sides of this old tank.
So it really depends on three variables; 1) the thickness and porosity of the rock, 2) its condition or where it's been since being left out to die, and 3) the maturity of the tank it's going into.
I and others have had some that lingered on and on, leaching bad stuff into the tank and preventing coral from growing for almost a year.
There is good news though. Read this thread, copied here from the Reefkeeping Tips (linked at the bottom of this post). Look for the part where I mention the product(s) that can shorten the cycle time for dry, dead LR.
Hope that helps. I would also suggest finding someone in your area that can give you some Macroalgae from their Refugium. Macroalgae can shorten and even completely eliminate the time it takes to ready a tank for animals. But maybe you've already read about that in the
Reefkeeping Tips thread.
"Keep your arms and hands in the tank and enjoy the ride." - L.A.Goon