As someone who just went through the process to qualify, it took FOUR WEEKS. I was warned from the outset it might take 2-4. Telling someone they wont know if their application for a single unit (the application isnt transferable) will be accepted for 2-4 weeks (when regular, non income restriced units take a few days to a week at most) makes it extremely difficult to shop around. How do you apply for a backup if you wont know if youll be approved for your first choice until maybe a month later?
There is a website run by the city to point you to these units, but it's one giant list/map, and has no knowledge if any of the units are actually available, just if a property participates. There are dozens (hundreds?) of properties with these units- it was a slog to find some available.