Thanks for the response!
The thing with Nest is that it's supposed to take into account the current indoor temp, outdoor temp, etc. to calculate how early it needs to start heating, and it also modulates the temperature of the water going through my convectors. So when it needs to heat a lot, it will go up to 75C. When it's reached the requested temperature, it will lower to 30C for example.
Looking back at this, this might be the reason it starts too soon. Could be that it thinks it needs to heat a lot and the burner is at 35-40C since it's not doing much, but then turns up the burner to 75C, which of course heats it quite fast.
I don't think there is a way to recreate such a system with home assistant.