It's not the temperature that they flee. Even tropical birds like parrots can easily fledge out to withstand very cold weather. Its the diminishing food supply. So birds who have the curiosity, cleverness, and ability to tolerate proximity to many other species, including humans, do just fine in the city, where the food supply never stops.
Out in the rural region where we live there are quite a few commercial aviaries. One fellow got up one morning and realized that we were having a once-in-a-century cold wave and the temperature had fallen below freezing. "Ohmigod!" he screamed, "My poor lorikeets will be frozen to death." He ran out in his jammies to his outdoor breeding colony and found one who had resolutely pecked a hole in the ice on top of his water dish... and was happily bathing in it.
A pair of grass keets escaped from the zoo in Stuttgart many years ago, where it gets REALLY cold in the winter. They were spotted in the spring, fat and happy.
People have macaws in Alaska and I've seen photos of them playing in the snow.