It's just a matter of removing the wires from the existing programmer and putting them in the terminals at the Nest that do the same function. You have posted a photo of the connections for your Danfoss programmer (I assume it is a programmer and not a programmable thermostat, as you don't actually tell us what it is) and you will have instructions for connecting your Nest Heatlink that came with it.
From your photos, the wiring at the existing backplane can be identified as:
N Neutral
L Live
1 Heating Common
3 not used
4 Heating On (Heating Call for Heat)
The wires from these terminals then need to be moved to the Nest terminals that carry out the same function:
N Neutral
L Live
2 Heating Common
3 Heating Call for Heat
Connection terminal 3 of the Danfoss programmer has no function, so it has just been used as a convenient connection for the Black & Grey wires. They will need to remain connected together by putting them in a separate screw terminal connector.
If you want to power the Nest Thermostat from the Heatlink, then T1 at the Thermostat goes to T1 at the Heatlink, and T2 at the Thermostat goes to T2 at the Heatlink. Or you can use an optional plug in power supply for the Nest Thermostat.
If you don't have a separate existing wired room thermostat, then that's job done.
However, Because the Nest carries out the functions of a thermostat, any existing thermostat should be removed. I suspect that the grey cable with Black / Grey / Brown wires does go to an existing thermostat. If you can confirm this, it will need removing from the wiring and the switching terminals bridged. This can be done in 3 ways, in order of preference, they are:
1. Remove the thermostat and wiring, and modify the wiring so that the terminals where the switched wires were connected are joined together.
2. Remove the existing thermostat and replace it with a junction box, connect the switching wires together. Isolate the neutral if there is one.
3. Leave the existing room thermostat in place and set it to the highest setting. (Or put the switching wires in the same terminal)
EDIT
If you can confirm the existence of a thermostat and provide details of it, I can advise how to take it out of the circuit.