Nest thermostat heating way beyond set temperature

if the nest is not detecting the W wire you have really three likely options

1. Something is wrong with the wire.
2. Something is wrong at the furnace
3. Something is wrong with the nest, which does happen.

If the old thermostat works then I would imagine that the controller units/boards etc in the furnace itself are PROBABLY ok, not a certainty but probably.

I would get a multimeter and check the output of W. Its possible that the voltage is too low for the nest to use, but the old thermostat works ok with it. With a lot of tech analog is more accommodating than its digital counterparts. Should be 24volt so if its testing near 24volts that might not be enough for the Nest.

It it is low then either the old furnace control boards/units are slowly degrading or there is poorly done splice. splice will need to be redone if you find it. could be anywhere between the furnace and the thermostat, start at the furnace end as most splices will be there. again you can test this with a multimeter, if you test the W coming out of the furnace and it reads 24v or more you know there is splice in the wall. if its less than 24v coming out of the furnace then that's your culprit and I would advise waiting for the new furnace.

You can also check the furnace and see if it is giving you any error code lights.

if none of the above, probably the nest. depending on who you bought it from, warrantys, and how long till you get the new furnace you might want to return it.

It is possible that adding a C wire would fix it.

From Nest support:

In some cases, you may experience problems after installing your Nest thermostat that can be fixed by connecting a C wire, or by removing the wire from the C connector.

• Heating or cooling is always on, and won’t turn off

This is just because without a C wire, the nest drains a little power through the other lines. It can wreck havic with some systems.

Otherwise, it could be that the Nest base is defective, you could contact support and inquire about them sending you a new one.

Did you put fresh batteries in the Nest? No. Because the Nest doesn't take batteries :)

So that raises a question. How does Nest power itself? It has a lithium battery (like a phone) but that has to be replenished somehow.

The answer is, Nest depends on the fact that thermostat wiring is in series with the furnace relay. Power flows

            -----R wire--------------------------- THERMOSTAT
         /                                           /
        /                                           /
       /                                           /
TRANSFORMER                    -------W wire-------
                             /
        ------C wire----- FURNACE RELAY 

Now, on most furnaces, the furnace relay is a physical relay. They require a certain amount of current flow to "pick up" or actuate. If current is too little, they will not pick up.

So if the thermostat leaked a small amount of current through the relay, it should not actuate. So the Nest takes advantage of that. It leaks that small amount of current - which typically will not actuate the relay on most furnaces - and uses that current flow to power itself. This is the same way old style dimmers work.

There are two ways this can blow up. #1 the relay may not be able to flow enough current to power the Nest. #2 that small current may be enough to actuate the relay and turn the furnace on. #2 is your problem.

Two ways to solve this.

Put a dummy load across the relay so current has a second path

            -----R wire--------------------------- THERMOSTAT
         /                                             /
        /       ---DUMMY LOAD--                       /
       /       /                                    /
TRANSFORMER   /                 >-----W wire---------
            /                 /
        ----+-C wire----- FURNACE RELAY 

Now current splits and follows 2 paths: through the dummy and also through the furnace relay. Smart sizing of the dummy load will divert enough current to stop the relay from actuating. On most furnaces.

Several companies make appropriate "dummy loads" for use with the Nest.

Add a "C" wire to the Nest

If none of these options work, Nest has the provision to take power directly off the transformer, by tapping the transformer via an extra wire: the "C" wire.

            -----R wire--------------------------- THERMOSTAT
         /                                          /     /
        /       -------------C wire-----------------     /
       /       /                                        /
TRANSFORMER   /                -------W wire------------
            /                /
        ----+-C wire----- FURNACE RELAY 

We haven't seen the sheath of your thermostat cable; typically furnace installers use larger cable than you immediately need and simply wrap the unneeded wires around the thermostat cable sheath. So you may have 1-3 more wires in that cable that are unused. If so, this is fairly easy.

A "wire multiplexer"

They make fairly complex powered gadgets which allow you to cram 5 wires worth of signal (R, W, G, Y, C) over 4 wires (or even 3). However they are a mighty octopus to wire, and prone to creating very difficult-to-troubleshoot problems, so I disrecommend them until the "dummy load" option has been exhausted.