Does the wire from the Nest go directly to the zone valve? That is usually how zone valves are wired and only one of the transformers is powering the Nest and the zone valve. The other one is for the boiler controls power.
A quick test is remove the Nest and remove the Red and White and connect them together.
Zone valve should open and boiler should start the circulator pump. See if buzzing goes away with them connected. If it goes away then you will need to have a C wire connected. If the transformer still buzzes then it could be going bad or the zone valve could be going bad. You can get a transformer at big box stores 24V 40VA is what you want they are usually in the plumbing heating dept near the baseboard stuff.
So if it was working good then this problem just started it could be the zone valve or the transformer going bad.
Red + starts at Transformer goes directly to Rh on the Nest (Thats your power in +)
White (- or Common) starts at Transformer and goes directly to Zone valve yellow wire (the common side yellow but they are not marked just 2 yellow wires)
Another White goes to the other yellow wire on zone valve (the positive yellow) and it goes directly to the W1 on the Nest.
So its a simple series circuit. Power Red to Nest, Nest wants heat and connects Rh to W1 like a switch, then power leaves the Nest on the White wire, goes to the Yellow zone valve makes the motor turn the valve and completes the circuit back to the - common side of the circuit at the transformer.
If you think you can run a new thermostat cable I would recommend using 5 conductor cable they sell it in 50ft and 100ft rolls at big box store. And pull it up the wall from the basement, if you strip outer brown insulation on old and new and fold them over and twist them with pliers they usually pull thru the wall easy.
Then you can connect a real C wire. Use blue and put it on the - side transformer screw with the white wire.
Houptee -- NJ Master HVAC Licensed Contractor
