You can send e-mails at home from your laptop using Windows Live Mail but not when you connect to the network outside home? So that's more of a network issue since you don't change anything in application's settings before you connect the laptop to a different network. It's possible that these networks block access to port 25 (TCP) that is used to connect to smtp.talktalk.net. Network administrators would do that in order to prevent spam from being sent from computers.
Alternatively, your firewall or antivirus software may prevent you from sending e-mails when you use wireless connection. You may want to try disabling them temporarily to see if it helps. If you use Windows Firewall then check
here for information on how to disable it.