I run into the "Briefly unavailable for scheduled maintenance" message regularly after updates. From other posts, to solve this, I've learned that you need to remove the ".maintenance" file from the root directory, but often times I load from my FTP client and no such file exists. I'm guessing this is a server/hosting issue as the server my client has his WP site hosted on is lousy and the updaters seem have a hard time removing the temporary files.
Anyway, this is what worked for me when I couldn't find the .maintenance file:
1. Using FTP client software, I duplicated the .htaccess file and renamed it ".maintenance" I did this in my local file directory.
2. Upload the new ".maintenance" file to your root WP directory (or the one with wp-config.php as other posts mention)
3. Delete the ".maintenance" file
I tested the site between #2 and #3 and it loaded, but had the contents of the htaccess file at the top of the page as that's what I'd put in the duplicate .maintenance file. You could also probably just upload an empty file called ".maintenance" and then delete it.
Hope this helps some folks!