The BTC buyer could click at home the payment sent button if they prefer to leave the laptop at home. To verify the receipt of the BTC a lookup at a blockexplorer for the receiving address (visible at the details window if you click on the trade ID) is sufficient to be sure that the trade has been completed once the seller confirms receipt.
But it would be probably easier if both traders bring their laptop and the buyer clicks the “payment started” button once handed over the Fiat. Then the seller should click the “payment received” button.
So in summary: Only the seller has to be online and click the “payment received” button if the buyer has clicked the “payment started” button at home already.
The buyer see a small text which indicates if the seller has received the message or if he was offline and the message was stored in his mailbox. Once he goes online he will receive the mailbox msg and processes it.
So it does not lead to a failed trade and it is a normal use case that the peer is offline during the trade protocol.
The mailbox is technically implemented that way that all peers store the encrypted message. Once the peer goes online he receives all pending mailbox messages and if tries to decrypt it, if he can decrypt it he knows it was his mailbox message and he goes on processing it and afterwards send a delete msg to the network so all nodes delete the mailbox message.