I have Windows 8.0. I use browsers to access my hotmail accounts via Outlook. But I have Windows Live Mail App set to the same accounts. I used to get a LockScreen update whenever I accumulated unread email on my main microsoft account. An ancillary account had an issue that forced me to have a new password sent. It seems since then I get no updates about unread mail on the LockScreen.
Live Mail is pinned to the Start Screen and I just checked that Live Tile is enabled. I have sent a couple of emails from the secondary account to the main one using a browser and I can see that the email has arrived. But when I look through the Windows Live Mail program it tells me there have been no messages received in the last two weeks.
I don't use Live Mail but it was nice when I looked over to the Lock Screen and saw I had a couple of unread emails. Now I have to manually check via browser.