As it isn’t your tree, you will be waiting for quite some time.
Just because you believe it’s dangerous (an arborist – in order to protect their ability to professionally insure themselves, also errs on the side of caution) doesn’t mean it has to be removed post haste.
Same applies even if it is listed as ‘weed’ within your LGA: it doesn’t mean the owners have to remove it post haste, as there are different classifications of weeds, and in some councils an established tree can be both a ‘weed’ and fall under the tree preservation act simply because of its age and size.
Assuming the owners corp actually wants their tree that is on their property removed, they will need to find funds (your offer to pay some of the costs notwithstanding). This can take up to 6 months.
I assume you are in NSW? If the tree is still standing after the freakish storms and 110+km winds that battered NSW for two days at the start of the month, then the tree is probably fine, which means that the owners corp may also have a harder time getting individual owners to agree to remove the tree. For individual owners, that tree may provide privacy, shade, etc, etc.
My honest advice would be to let it rest and save yourself some angst as they are under no obligation to remove a tree on their property.