Yoiur account with BT Yahoo is most likely a POP account, meaning that when you receive a message (like with Apple's Mail program), the message is downloaded to your computer and removed from the server (hence, why you cannot access those messages via webmail -- they've been downloaded to your Mac and removed from the server).
In your account settings on Apple Mail, there are options in your BT Yahoo account to "leave messages on the server" or something similar to that. I'm not in front of a Mac right now, so I can't give you specific steps to take, but will be in about 8 hours and can post back with how to make the messages "stay" on the server so you can access them remotely.