For the boiler to operate and yet not provide heat to either the heating or hot water there are only two possibilities:when you press the button on the heat link the boiler turns on however it doesn't turn the heating or hot water on so im guessing I have either missed a link or connection.
a. either the existing system has never worked, or
b. you have done far more than just remove the programmer and fit the Nest instead.
The electrical side of basic heating systems is just a set of switches, and really isn't complex or difficult at all.Clearly heating systems are more than just wiring though!