First of all there is something odd here. I'm guessing the existing thermostat isn't working as the wires in your photo are in the wrong terminals. They are in N/C and Com, but......
........ they should be in Com and N/O. Do you have an explanation for this? It would be good to hear it before you proceed. But If you don't know, I suggest you get a pro in to check it out and find out what's happening at the other end of the cable and what it's connected to. Wired as it stands the heating would normally be working in reverse. 'On' when it should be 'Off', and 'Off' when it should be 'On'.
Also, you don't specify which version of the ecoFit Pure Vaillant boiler that you have. As you don't mention any hot water controls I assume that it's a 'combi boiler' and not a 'heat only boiler' with a separate hot water cylinder that you also wish to connect to the Nest.
Assuming it's confirmed the wires were originally in N/O and Com, if you have a Nest-e which is also battery operated like the existing thermostat then the wires would go to terminals C and N/O. It doesn't matter which way around, but convention has it that the brown wire is C.
Danfoss N/O = Nest-e N/O
Danfoss Com = Nest-e C
However, if instead you have a 3rd Gen Nest
Then the wires go Heating Common, terminal 2 and Heating Call for Heat terminal 3 [aka normally open or N/O]
Danfoss N/O = Nest (3) Heating Call for Heat [normally open or n/o]
Danfoss Com = Nest (2) Heating Common
However this version of Nest is not battery operated so it also needs a 230V mains supply connecting to N & L to provide it with operating power. This mains supply needs to come from the same 3A fused mains supply as the boiler.