I'm probably going to be the bearer of unwelcome news in advising that TalkTalk don't offer a backup service for lost or permanently deleted mail messages.
If you've just deleted the messages from a folder then TalkTalk Mail moves deleted messages to the Trash folder. But if you've emptied the Trash folder then that is a permanent deletion and cannot be recovered.
Your options are to see if you personally made backup copies or perhaps downloaded mail to a device connecting as a POP3 account that stores copies in the POP3 Inbox just on that device. Ask originators of mail messages if they kept copies or if recipients of your messages kept copies. Perhaps look in the Sent folder to see if you have copies of messages you sent that include a string of conversations and replies.
The TalkTalk advice is always to make your own regular backups of wanted mail messages.
