I speak from experience... sponge filters ,run in an est. tank, are going to take at least 3-4 weeks to "seed"enough to run a tank on their own ,if you want to use another filter from one of your est. tanks(providing the filter is established)this will greatly increase cycling speed..
This is correct.
Fresh de-chlorinated water in a new tank will contain only the bacteria that land on the water surface from the air (although some believe that they may exist in small quantities from the tap water itself.) They are non motile (no spores) and will only reach the sponge randomly unless a full population is adjacent to the sponge. Consider the demographics of the BB population: it will only double every 7-20 hours (it will be more at the upper end unless conditions are optimal for temp, food, pH, surface structure.)
BB bacteria exist in densities of at least 1 billion per milliliter (.061 cubic inches) for nitrosomonas and 10,000,000 for nitrobacter (see
http://www.ceric.net/wonmun3/ksee/04708383.pdf). Please note that this count for nitrobacter is likely low by a factor of 100 since the study was done in a high ammonia, low nitrite environment.
A small, mature sponge filter would have hundreds times this figure. Assuming a random start of 100 nitrosomonas and a 16 hour double rate, it would take 3 weeks to reach 214 billion of the needed BB. Roughly speaking, that's like 13 cubic inches of mature BB.
Starting with a seeded sponge or material from a seeded sponge is the way to go.