Home has separate thermostats for AC and hot water heating

We recently moved to a new home (new to us; not a new build). The AC in our house uses a separate system/wiring from the heat. The AC thermostat is upstairs and the one controlling the heat is on the main floor. Both currently have smart thermostats with their own apps.

My question is this: if I replaced each system with a Nest, would they integrate together and work in unison? Could I control heating a cooling on either thermostat and do things like set a temperature range?

For example, with the air on it is always 0.5-1c cooler upstairs.