I’ve determined that the problem only occurs for me in Chrome when I have Abine’s DoNotTrackMe extension enabled.
I just updated Jetpack to 2.2.5 and created a new Twitter widget for this purpose, and am also getting a blank timeline. The old widget still displays tweets. I also tried to use the shortcode in a new post, and the shortcode was not processed (the shortcode was output onto the post). This is the entire output from the Jetpack Twitter Timeline widget:
<li id="twitter_timeline-2" class="widget-container widget_twitter_timeline"><h3 class="widget-title">Twitter Updates</h3><a class="twitter-timeline" data-widget-id="[my id]" data-theme="light" data-link-color="#f96e5b" data-border-color="#e8e8e8" data-chrome="noheader nofooter noborders transparent" width="200" height="400" lang="EN">My Tweets</a></li>