[Simon..Hewison]

From Tesla's Gen 3 wall connector's installation instructions:

https://www.tesla.com/sites/default/files/support/charging/Gen_3_Wall_Connector_Manual_UK.pdf

Additional requirements apply to UK properties fed from a TN-C-S supply. In most scenarios, these
properties will require an external device which provides PEN fault detection and isolation in accordance
with BS 7671. It is the responsibility of the installer to ensure that the requirements of BS7671 are met -
please refer to Tesla's application note at the end of this manual that discusses this topic in more detail.

So, no, Tesla's kit doesn't do PEN fault detection itself, there would have to be another solution to address the rare cases when the supplier's TN-C-S earth fails. An external box like a Matt-E would work, as should a dedicated local earth rod (but for some reason, Tesla don't think it would, presumably because a local earth rod might not be as easy to install and get a good earth as the installer would hope it to be.)
Any qualified EVSE installer electrician will be able to advise what would suit your installation.