There are two ways to determine if a forest is regrown. First, the trees can reach approximately their maximum height. Second, the first generation of trees can die, leaving lots of fallen decayed wood and a second generation of trees, that might be of a different species (this is called forest succession) takes over.
I don't know many details about European trees, but I can tell you a lot about forests of Eastern North America.
Lake Anna state park in Virginia was clear cut in the 1920s. It was logging land from then into the 1970s, so it was planted in pine and cut every ~25 years. In 1971, the lake (Lake Anna, believe it or not) in the park was used as cooling for a nuclear power plant, so the land was turned into a state park.
Originally, the land (alluvial bottomland on the North Anna river) was covered with fire resistant oak-hickory forest. That was all removed by the 1920, and pine was planted. Since the forest has become a protected park, the early pine has given way in succession to tulip-tree, red maple, and sweetgum, mixed with some remaining shortleaf pine. Pines don't live that long, and they don't grow as fast as the hardwoods; so despite a ~20 year head start the tulip-tree and sweetgum in particular are already as tall as the tallest pines, perhaps 25 meters maximum after ~50 years of growth (the maples are probably only 15 meters, maximum).
So in the one sense, the forest is fully regrown in 50 years. On the other hand, the first generation of hardwoods have not died yet. Tulip-tree and sweetgum are both shade intolerant, so they will not produce a second generation in the same forests that currently exist. Instead, the forest will be replaced by beech and maples.
However, there is still another twist. The state forest service removes fallen wood and prevents wildfires. But Virginia sees regular summer droughts, where high temps make for fire conditions. There would normally be understory fires every 5 years or so, under pure natural conditions. With these fires, beech and maples will have difficulty establishing. Instead, white oak and shagbark hickory will grow; these are not as shade tolerant, but are fire resistant as young trees.
What would happen in Virginia, from a bare field
In cleared fields, there isn't much danger of fire, since there isn't much material to burn. A variety of shrubby trees will start growing. After about 50 years, tulip-tree and sweetgum will have 'won' the race. They will form a closed canopy about 25 meters in height. In the understory, shade tolerant red maple will dominate, along with shrubbier trees like holly, dogwood and persimmon. After this time, the amount of tree litter will build up to the point that understory fires every five years will prevent trees from being established that are not fire resistant.
The only young trees that will survive are oaks and hickories, particularly white oak. After 100 years, the first white oaks will reach the 25 meter canopy. Some of the Tulip-trees (the tallest hardwood in Eastern US) will have reached 50 meters. The short-lived red maples will start falling at this point, and due to fires they will not be replaced.
After 200 years, the tulip-tree and sweetgum will start dying; the only remaining specimens will be the biggest ones. After 400 years, all the tulip-tree and sweetgum will be gone, the first generation of hickory will have fallen and even the first generation of oaks will start falling as well. At this point you will finally have an old growth forest, which will change little over time.
50 years for tall trees and a closed canopy. 100 years for succession to final forest type. 400 years for an old-growth forest.
Of course, this is for Virginia. You could do an analysis for any region with its specific forest type. But the numbers here are reasonable for any temperate forest.