How do I know if my hot water boiler is compatible with Nests domestic hot water control
Unless you already have a separate electronic controller/programmer for your water heater you can't control it with your Nest thermostat. Since you haven't mentioned it or provided pictures of a controller or wiring, I assume you don't have one. Having a controller for your water heater is common in some regions but not in others. Based on search results from Nest's UK website, I guess it's more common in Europe. In the southern US, I've never seen a controllable water heater.
A basic water heater is controlled by a knob or adjustment screw and doesn't contain any electronics to interface with.
I take it that in the box there are 2 x "Heat Links" (HL for short.)
One will replace the HW (Hot Water) thermostat and the other is for the room thermostat CH (Heating)
If your boiler is "Open Therm" compatible it will most likely work with Nest, if not "Open Therm" don't worry (yet)as it may still work.