When you switch to TalkTalk as your mobile phone provider, by default Visual Voicemail for your iPhone is not enabled.
And you're stuck with the 90s voiceprompt of the 'Talkbox'.

The following steps will activate Visual Voicemail for your iPhone:

1. Send a SMS text message with VVM ON to the number 935.
2. Shortly after you should get a text message confirming that Visual Voicemail has been enabled for you.
3. Now on the iPhone, open the phone app and go to the voicemail tab. There you will either see a button asking you to setup the voicemail or a button taking you to the 90s voiceprompt.
4. Do click on this button and setup the six-digit PIN code for your voicemail (this can be done either via a call/voiceprompt or via the guided iPhone button/dialog).
5. Once you have setup the PIN for your voicemail, close the phone app on your iPhone.
6. Then open the phone app again and go to the voicemail tab, where it should show the usual Visual Voicemail list of missed calls and no longer the button to call the 90s voiceprompt.
   In my case it took a couple minutes for this to work, thus some patience might be needed.
7. Congratulations, you have now a working Visual Voicemail on your iPhone with TalkTalk :-)