A simple way to check a bunch of URLs for mobile-friendliness is to use the PageSpeed insights API. Google just launched an update to the API that gives you a bit more information about speed & score, as well as a clearer format for text-responses. It is announced on Google+ by John Muller.
You can check it out athttps://developers.google.com/speed/docs/insights/v2/getting-started
Just grab an API key, copy & paste the code samples (or use the command line, if you’re into that), and maybe even get started on creating a physical speed meter for your homepage ( http://www.jacobquatier.com/posts/2014/12/01/the-meter/) as a weekend project. If you have a bunch of clients that you’re watching out for, it won’t take much to track their important pages like this automatically.