Okay, after some playing around, I figured it out. The file size of the GEDCOM was just too big. I had to increase the max upload filesize in my php.ini file and that solved the issue.
Thanks for the clarification, Kreg.
I figured it out! I had accidentally deleted the code <?php wp_footer(); ?> from the footer.php file in my /twentyeleven theme folder at some point. All I had to do was reinsert <?php wp_footer(); ?> into that file and now the Speakup plugin works. Not entirely sure why that was the case, but for anyone else who ever has this problem, that’s the solution.