Let's start from basics. TalkTalk Mail can be accessed on a laptop, desktop or mobile device via an Internet browser like Edge, Firefox or Chrome. So all you need is an up to date Internet browser that will already be on the laptop and phone.
Select here: Sign in to TalkTalk Mail
Enter your full TalkTalk Mail tiscali email address and tiscali password, select Sign in.
You will see the exact same Inbox although the presentation in a desktop / laptop / notebook browser is different from a mobile device that has a smaller screen area and usually portrait view whereas the laptop is landscape view.
Email apps and email software can be set up to view the exact same Inbox and folder content if setup as an IMAP email account. If you use on the laptop email software like Thunderbird or Bluemail and an email app like the Gmail app or BlueMail app on the phone then that also should be setup as an IMAP email account to make sure the folders and Inbox are identical in terms of their content.
Do NOT use the Tiscali Italia app as that's totally unrelated to tiscali.co.uk managed by TalkTalk.
Now, when using an Internet browser to access your mailbox on the laptop if the Chat transcript is not there in the Inbox then it's been downloaded from the Inbox and the copy that did arrive has been deleted. That happens if the laptop is using email software set up as a POP3 email account using default settings. Because the Chat transcript has been downloaded to an Inbox that's part of the email software on the laptop then you see it on the laptop but it's not available to see on the phone regardless of whether you're using an Internet browser or email app.
To check this out perhaps you'd tell me exactly how you access mail on the laptop, what operating system is running on the laptop and what email software is in use and its version number. And then whether you set up an IMAP email account or POP3 email account?
