There will not be an evacuation, since we don't usually know space rocks are coming. It has only happened a couple of times we've seen a rock before it hits the Earth (with all of them being small and burning up in the atmosphere). One time I got an alert because the thing was coming down near me (and I'm a total nerd being subscribed to such a thing), I went outside and looked up and even though it was clouding I saw a big shooting star.

We've mapped a lot of stuff in the solar system, so we can be fairly sure anything big coming near Earth isn't going to hit in the coming decades. But there can always be something we miss, something large enough to take out a city. We are getting better at spotting it, but lead times are hours not days or weeks.

If you are talking about evacuating a city, that's a tall order in a couple of days in the best conditions. Evacuating an entire country (small country, say 15-25 million people), I don't know if that's even possible, that's never been done.

After everything is said and done, recovery will take years. Something that big will cause a mini ice age with lower temperatures for a couple of years, this wrecks food production around the world. We've learned the last 10 years how vulnerable our logistical processes are. Something like a big meteor impact will be much much worse. https://en.wikipedia.org/wiki/Little_Ice_Age

Because it takes say 20-30 years to get to some kind of recovery, most people will have to live somewhere else and get rooted there in that time. But knowing humans, we will probably claw back the land. Look at Hiroshima for example, completely wasted, now a beautiful city. Something on the scale of an entire country has never been done, but it can be done. New people would settle the land during the recovery, but most likely the original culture and ethnicity gets diluted and maybe lost with people spreading out.

Depending on what exactly happens there will be less refugees than say for example a big war. If a big city is hit directly, it can easily kill everyone in the city, comparable to a nuclear strike. It hard to say, because if the meteor is big enough to destroy the entire country, that's an ELE. Most likely there's something that destroys like a city and the rest of the country will have issues with the fallout, but that's manageable.

It also really depends on where the strike hits, if it's a country like Australia with densely populated cities and a whole lot of nothing, a strike can kill a lot of people and leave the country relatively unaffected or even hit and kill nearly nobody. If the strike is in for example Luxembourg, the country simply stops existing and every inhabitant is dead.