Short answer: It depends.
Longer answer: What sort of glacier was it, at what latitude and altitude, what is the regional windflow pattern, if such has been determined, and at what rate did the ice retreat?
Now that that’s settled . . .
What did happen at the end of the last Ice Age, and what were the results? I’m not going to worry about the whys of the end of the Wisconsonian (North American term) glaciation around 14,000 years ago, but will look at the effects and what we think happened as the ice melted. I say “think” because it is hard to tell in some cases, since 14,000 years of erosion and climate shifts have happened since, and because some of the results are . . . seriously odd and may be either hyper localized or the result of test sample contamination.
So, let’s say you are standing where a continental, as opposed to mountain, glacier has just been. It is six months since the ice left. You see a rough, hummocky landscape of rocks and dirt, with boggy spots and lakes of varying size all over the place. The finest dirt blows past, although it is not as bad as when the ice was closer and everything was dryer. Loess blows worse than sand, and can form super dunes, or hills, several hundred feet tall, like the Loess Hills along the Missouri River in Iowa. You may find a few mosses and lichen, but not much plant life yet, because of the cold wind coming off the glacier ten miles or so to the north. Ten miles? Yes, it appears some glaciers retreated at a gallop as well as advancing at a gallop. Once melting started, it accelerated as more dark surfaces were uncovered, allowing the sun to heat more of the surface, which allowed things to warm more, which melted more ice, which exposed more soil . . .
One thing to keep in mind is that as the continent-size glaciers retreated, they put a lot of water into the air, and affected weather. Think about Minnesota, Wisconsin, and the Dakotas and northern Iowa. They were left marshes because of the melting ice and still are in some areas (the “pothole prairies”). The glaciers had reworked old rivers, leaving an enormous swath of what is now the US and Canada without drainage. It was one enormous marsh, with a few rivers. Likewise northern Europe, which is why so much is so flat. So you will get lichen and mosses, then cold-hardy plants that are probably spread by the wind or birds. They provide a start, and at least in North America, grasses followed, then trees raced north from more protected areas. It looks as if, in a few cases, once the retreat really got going, the tundra-taiga-grassland-pine forest transition happened in as little as a century or so. This was in areas that had enough moisture. Drier areas, like the Dakotas, stayed grassland, a process later helped by deliberate burning.
And those rivers? Humongous. Bigger than the Amazon, Nile, Danube, and Mississippi combined, at least for short periods. Once that water started flowing, it flowed. The gigantic lake that geologists call Lake Bonniville, and its associates Glacial Lake Missoula and Glacial Lake Agassiz all had spells where ice dams would form, holding back the water, then get undercut. Walls of water ripped across the landscape, creating things like the Channeled Scablands* in Oregon, or just over-sized valleys in Minnesota and South Dakota.
Note that the farther north your glacier was, or the higher in elevation, the slower the process happened. Even so, the case of the rivers of what is now Punjab is useful. They came from the Himalaya, and were rather impressive during the Ice Age. Beds twenty kilometers wide, rivers that flooded from the mountains through what is now a desert and reached the Indian Ocean . . . Huge. And then around 3500 BC/BCE or so something happened. One, the melting slowed and two, an earthquake diverted one of the rivers. This combination led to shifts in flows, reductions in flows, and to lost cities where once mighty rivers had run, and legends of the Lost Sarasvati River. Speed and scale matter, as does geology. Oh, and keep in mind that as the ice melts, the land bounces up in isostatic rebound. That will change what can grow where, while exposing some land that would otherwise have been submerged. You can see “bathtub ring” layers on the shore of Lake Superior that are the result of erosion as the land rose.
In other places, away from the ice, you got some fascinating blends of things. To use the Texas Panhandle as an example, during the Ice Ages, the region was a temperate savannah, with trees, brush, and grass, far more plentiful water, and occasional visits by large floods from the growing Rocky Mountains. Think Africa, but with woolly mammoths and rhinos, giant sloths, dire wolves, saber-toothed cats and other predators, armadillos the size of a VW Bug, and other critters. As the climate warmed, the shift in timing and range of precipitation, and the widening range of temperatures, made life difficult for the plants and animals. The megafauna died out along with the trees and leafy bushes, replaced over a few centuries by bunchgrasses, C-4 plants that could handle the -10F to 100F temperature range and swings of aridity. The region stayed relatively more humid than it is today until about 12000 years ago, when periods of aridity and humidity alternated, hitting a low point in terms of moisture around 5000-3000 BC/BCE. Some pines and other “mountain” plants may have lingered in the Canadian Breaks and other refugia (protected areas) through the Folsom times (10,000 BC/BCE) but it is hard to tell, because the pollen could have blown in from the west rather than being local.
Blowing pollen is one of the big problems for identifying how fast and how the biotic community changed as the glaciers retreated. We (historians and archaeologists and other people who dig old bones, literally as well as metaphorically) use a lot of pollen studies for reconstructing environments when we don’t have bones, charcoal, amber, and other things to use. But pollen blows, it can look like other pollen, it can wash into ponds (not as much of a problem), it can be transported by waterfowl and insects…
And we can be surprised. For a long time it was assumed that northern Germany had been uninhabited during the last ice age, with no humans able to deal with the harsh climate. Surprise! Starting in the 1930s, and then really getting in gear in the 1960s-70s, it turns out that paleolithic and neolithic peoples did indeed live up there, or at least passed through and left tools, bones of things they ate, and other evidence. Were the conditions not as harsh as people thought? Are humans just that persistent and resilient?
Where can you learn more? It’s a little tricky, because so much of this research is done in Europe and published in European journals, often in German or French. The work in English tends to be in academic articles or conference papers. However . . . Several of the Roadside Geology of [state or park] volumes talk about early Holocene climate and changes. Minnesota Underfoot! has good material.
TL;DR – It depends on where you are and when, and varied a great deal.
Paul Sears wrote a very early article that presents what was known and how it was known in the early 1930s: http://people.wku.edu/charles.smith/biogeog/SEAR1935.htm
Here is the abstract of a later article that gives you a sense of what is out there. https://www.researchgate.net/publication/290792865_Some_aspects_of_European_late-_and_post-glacial_flora_history
A little from Glacier NP about what they have, pointing out that they are post-glacial. https://www.nps.gov/glac/learn/nature/plants.htm
The Remembered Land by Leary and Hodges is a good read about archaeology and post-ice age environments in Britain.
E.C. Pielou’s After the Ice Age is another general overview focusing on North America. It is a little dated, and is a translation, but is readable and useful.
Paul S. Martin’s two edited paper collections, Pleistocene Extinctions: The Search for a Cause and Quaternary Extinctions: A Prehistoric Revolution focus on the demise of the megafauna (wooly mammoth et al) but the second one also has some really interesting climate change work.
Likewise Riversleigh by Michael Archer is about the paleontology of ice-age and later Australia and what happened to the critters there. As well as what happens to geologists in the field in the Back of Burke, out past the black stump.
Looking for books on Late Pleistocene ecology, or early Holocene, generates a number of paper collections on Amazon.
Oh, almost forgot! Glacial Lake Missoula and its Humongous Floods You can guess the subject.
*Pssst, Dot? If you need another “looming disaster, will they take heed” event…