So I have had some success getting a Wordpress website running as a child of an umbraco parent.
I created a Wordpress site, using Azure and then copied that site to a child /blog dir successfully using this article: http://blogs.msdn.com/b/webapps/archive/2013/02/13/hosting-wordpress-in-a-subfolder-of-your-windows-azure-web-site.aspx
You can see my success here: http://littlenomads.azurewebsites.net/blog/
Part of the solution so far was to change the Umbraco web.config to ignore the blog directory: following this SO answer: PHP configuration on .NET server - URL redirect
So this means if the Wordpress blog is loading the MySQL database connection is working.
But when I try and login to wp-admin, the website spins for a while and then I am getting a 502 error which means Azure is crashing waiting for a long running request I believe.
I have tried debugging following this guide: http://ruslany.net/2013/01/php-troubleshooting-in-windows-azure-web-sites/
I am seeing failed request errors for wp-login.php but this does give me any direction. I am not seeing any PHP errors. http://littlenomads.azurewebsites.net/blog/wp-login.php
I am just not sure what the issue is and have no debugging direction at this point.