LocalBitcoins only handles the bitcoin portion of the sale they do not and can not know the status of the bank payments, cash transfers, or anything that isn't the movement of bitcoins.
Escrow is there to protect the buyer, so they can pay the seller with the assurance that the seller must deliver the bitcoins if payment is rendered.
As the bitcoin seller, it's up to you to make sure you actually get paid - LocalBitcoins can't take care of that for you.
Once you've been paid you release the bitcoins to the buyer using the button on the web site. The bitcoins will then instantly show up in the buyer's LocalBitcoins wallet.