Pretty sure I have the latest MFT and just updated my iPhone 5 to 6.1 about an hour ago and saw this post and figured i'd try it since I could never get it to work before and I discovered some stuff that "I" didn't know before.
When you pair the phone you have to go into the phones bluetooth settings under synch and enable Show Notifications. Didn't know that was before.
Doing some testing to see if I could get it to work I noticed something. For incoming messages I have my Notifications for "Messages" set to NOT show on the lock screen and NO banner. Just to show in the notification center and the badge icon. With those settings MFT wouldn't recognize any messages. If I turned on banners it made no difference. If I turned on the "Show on lock screen" it actually made it work. With that setting enabled messages would appear but the "Reply" box is grayed out. For iMessages it isn't grayed out (No idea why) but it won't send a message so I assume this is exactly how it behaved before.
'12 Sterling Grey Titanium Hatch - Moonroof / Winter Package / DCT / 17' Alloy Wheels
MK3 Manual Transmission Swap: http://bit.ly/1ondtuV