This is what I have at the furnace. There is no Y on the furnace but Y is hooked up at the existing thermostat. There is also a c at the thermostat not hooked up (current thermostat is battery). This is at my parents house. I am not there at the moment but I can go back and forth.
I assume that smaller grey sleeve of wires controls the central air
I used a small piece of wire and all I did was went from the c on the control to the c on the thermostat to see if it would power on and it didn’t. I have no idea if it works that way for a test
One more thing. In the second picture, the wire sleeve on the left goes to the furnace, the one on the right goes to the thermostat
Please share current thermostat hookup picture. Is the white wire not connected to anything? Does this house have heat?
From images you have already provided, I think you just need to add the blue wire that runs to the thermostat to the blue and white wire-nut, and your wyze should power up. Also assuming you want heat to work, connect the white wire from the furnace to the thermostat white wire, assuming the white wire in the thermostat is connected to W1