thermostat connections with different size wires
Do make 100% certain that the 14AWG wires are not line voltage. There are such things as line voltage thermostats that are designed to directly switch the mains voltage to a device such as a heater or fan unit. It would generally be unusual to find a low voltage thermostat connected up with a large sized wire unless it was a line voltage thermostat. 

If indeed the wires are verified to be low voltage 24VAC then you can for sure use the properly sized twist on wire nut to join 18AWG to 14AWG wire. 

If there is any feasibility to replace the 14AWG wire used for a low voltage application like this with widely used multi conductor thermostat wire please do consider doing so. It will help to eliminate future confusion. At the same time as replacing do use wire with more than two conductors so that there is flexibility in the future to use a smart thermostat, or one with additional fan and cooling controls.