First, make sure that your server responds to requests for the old, original URL with a 301 redirect to the new SEO-friendly URL. Then go into Webmaster Tools and use the Fetch as Googlebot tool to fetch the old URL. Once Googlebot has fetched the URL, select "Submit to Index".
It is hard to say. But it will be indexed within 10-60 days. When you have changed your url then you should share that link to the social media sites and also social bookmarking sites to get it indexed quickly.
Generally, a website will be crawled by the google search engine bot based on the no. of visitors for that website and the specific timeperiod for those webistes will be based on the traffic level of each website.