As I understand Google will visit/crawl your website more or less immediately when you apply "Fetch as Google".. However that does not mean the changes will be shown on the search results immediately... further once Google has crawled your site there is nothing else you can do but to wait for when the data is displayed within the search results... In addition this covers both mobile and desktop... Keep in mind mobile is just one factor out of many others which G examines...
Personally I never bother with using the "fetch"... In fact for a lot of my sites such as PBNs - I do not use Google tools such as WMT, Analytics etc in order to minimize footprints etc... I really do not concern myself when G will visit... You can be sure of one thing they will visit...
BTW - you can check the last G visit date by checking your server log files or by clicking on the green down arrow next to the website URL and selecting "cached" which provides the last date/time of crawl and how the website then looked....
In my opinion a better way to get G crawling your site is when you create new backlinks to it... e.g. forum sig link, youtube video, guest article etc... Those crawlers will come rushing in...
Hope that helps! Best of luck ;-)