How can I speed up the module registry approval proces on Drupal.org?
The process can be very slow unfortunately. There probably isn't a complete answer, but I can give a few hints.

Try to find users which test and report that it works (as detailed as possible), developers to review it and report that too. There are many ways to do that, for example for developers:


Ask in IRC (#drupal-contribute) (Remember to offer something in return, e.g. patch reviews). Do not ask multiple times and only when the channel is quiet.
Review/test applications of other users and ask for a review of yours in return.
If your module interacts with other modules, ask their maintainers to review the integration code. Again, you could help with the issue queue of these modules (review, write patches, answer support questions)


This has two advantages. a) It shows that your module works and users are interested in using it. b) It helps to keep it on top of the (very long) list of issues.

Also, try to make it as easy as possible to install and test your module. For example, if your module integrates with a third-party service/website, provide a test account for it or make it possible to register there for free (if it isn't anyway) and provide detailed installation instructions (for example in a README.txt or INSTALL.txt file).