It could happen quickly if the second bank says "yes" then it will go pretty quickly. Also, the home may no longer be available for a short sale if the bank files a foreclosure on the property. Then the wheels are in motion, and they do not reverse that decision lightly and usually not at all. When they file the foreclosure, the attorneys take over, and they proceed until the bank is declared the owner. Then the bank will once again offer the home for sale. If you have been told you have a deadline of Sept 29th, they probably mean what they say. This is all such uncharted territory that we cannot be sure, but that deadline should not be simply ignored.
The first and second mortgage holders will work out something because it is in their best interests to do so. Something is better than nothing. I have know of second mortgage holders who have bought out the first mortgage to avoid losing more than they feel they will if the own and then sell the home themselves. Foreclosures are so plentiful that there are some generalities but some exceptions too.