Submitting a disavow file is dissimilar to sending a reconsideration request. It is an automated filter acting between your site and the inbound links. A little like sun tan lotion. It filters out (NOFOLLOWs) the links as they come in.
It inherently works in a progressive manner as the toxic effect of the links are only removed when the spiders re-crawl the pages with the links on them, eventually follows the links and then, if the links are on the disavow file, they get filtered out at that point.
So, in answer to your question, the file can start working almost immediately or it could take a very long time depending on the scale of the file.
However, typically, people start seeing some movement in rankings (assuming it is done correctly) within 2 or 3 weeks when a significant section of the spammy pages starts to get re-crawled.