Typically the temperature difference between day and night is 10 C. So at first approximation, it drops 1 C/hour.
So if you started at a middle of the road temp, of 20 C, you'd be down to freezing in a day.
As the temp drops, clouds form. This slows down the heat loss to something like half to 1/3 of the previous rate.
Since it doesn't rain every night, it takes more than a day for this to happen. So continueing with the furious handwaving: Temp drops 1 C/hour for 10 hours. Clouds form, and it drops for 1/2 C until it gets to freezing. So at 30 hours we are at freezing.
At this point another effect comes in: Latent heat of water Cooling is going to essentially stop while the top foot of ground freezes, and most of the moisture comes out of the atmosphere. This will take at least two full days, more in wet climates (more water in the soil) So at the end of day 3 we are at something like -5 at temperate latitudes.
Now we get some differential effects. Areas near very large lakes and oceans will be warmed by the these bodies of water that take a long time to cool. However continential interiors will continue to cool. Once the air is dry, it will go back to that 1 C/hour, -- call it 20 C/day. The air itself won't chill that fast, and you will end up with layers of very cold dense air near ground level with a temperature inversion above that. This may foster cloud formation again, and limit the heat loss. So, wind it back to 10C/day.
That cold air will flow down hill. Coasts will have a land breeze of cold air flowing off the continent onto the ocean. Oceans will heat it up and humid air will move inland somewhat, producing heavy snow.
At some point the oceans start to freeze at the edges. Places that have ocean currents off shore will last longer. Bays, lakes, and places isolated from currents will freeze as soon as the water temp drops to zero. This is largely a function of water depth. In Alberta it takes a good month of cold weather before even a 10 foot deep slough is safe for skating (4" thick ice)
There is enough heat transfer between equator and poles that even during a 6 month full dark at the south pole temps only get down to the -120s F (-85 C)
Much of that will be the thermal inertia of the ice itself -- which isn't much better than rock. (Ice has a specific heat half that of liquid water)
So overall: Killing frosts. This would likely kill most land based plants (Even plants that can survive a -50 winter can't do so if it catches them during the growing season. They need to adapt slowly.