The iOS 6.1 update fixes many huge bugs like the wifi and no service but it has introduced a new one that stops the loudspeaker from working with iOS 6.1. It doesn’t impact everyone but there is a workaround, well sort of.
Firstly you might find that not all audio is crippled. Voice Memo app may well work ok and you will also be able to hear your phone ringing and the speakerphone will function fine during the conversation, but you’ll get no sound from a music track, video file or game.
Restarting the iPhone doesn’t work. So what do you do? and where is the giveaway sign there is a problem. If you have access to AirPlay, you’ll notice something odd that the choice offered to you is usually the local AppleTV and your iPhone, but now the iPhone option alternates between it and Dock Connector. This means that your iPhone believes it’s connected to a Dock and insists on rerouting the sound to a non-existent extension.
So all you can do is restore from fresh. This will wipe your phone and data. If this doesn’t work it’s a trip to the Apple store or Apple support. Not ideal, but then it’s not affecting many people either.