First, make sure that during the upload process the URL format option is set to "File URL" instead of "Attachment Post URL." WordPress should automatically detect the presence of an MP3 file attachment and display the native player.
If WP does not display the native player even when it knows it is an MP3, something in your theme or other plugins is conflicting with the native player. There are too many possibilities to really know why that would be happening. I suggest trying the .htaccess hack described in this thread as a last resort.
Alternatively, if your native player refuses to work, some users have just switched to a 3rd party player like mp.miniaudioplayer.