Wiring Nest Thermostat E Heatlink in place of original Thermostat
Google seems to be truly unhelpful on this one of their products.

Your existing thermostat has a live feed in at terminal 1, which is brown (as it should be) and the neutral is connected to terminal 2 (it should be blue, and is sleeved to indicate so), that is likely there for a function light. The switched output is at terminal 3, looks like that's a black wire.

So you need the brown to the C terminal and the black to NO.

Here's the only image I could find, from here



The E version is battery operated, so doesn't use any mains power. No ground connection or neutral are needed to the Heat Link, so those wires should be insulated and not connected.