Can't speak for Europe but The American left was not dependent on Communism in the first place, in fact they went top great lengths to not be Communist during the Cold War, and often had to shoot themselves in the foot politically by going out of their way to not appeal to the socialist factions of their own wing of the poliutical spectrum, resulting in it taking uniquely powerful leaders in order to advance their vision at all (LBJ, JFK).
The demise of the massive international leftist bloc was a Godsend for the American left IMHO. After a very brief surge of patriotic fervor that saw George HW Bush elected to a mediocre term as President, the nation actually turned inward for most of the 2000s, which is a perfect time for leftists to advance social policies, and we indeed saw many adopted.
The first decade of the 21st century actually saw a lot of environmental reforms go through in particular, because the green left was free to tap the red left for additional help and organization to advance their agendas for the first time without torpedoing their political progress. Initially the results were clumsy, with the old Socialist organs acting way too much like old Socialist organs, but that wore off quickly, and now left wing academia has been a powerful force of the political left for a couple generations and they're getting more savvy in their operations.
This was the era where the liberal entrenchment in academia became semipermanent and a lot of the old guard left began to switch from a kind of institutional or national leftism focused on economics and institutional changes, to a more individually focused version of sociological leftism. Whatever your feelings of the movement as a whole, Social Justice, while present before this era in the liberal platform, started to really take the foreground in this period, so did so-called Third Wave Feminism,
TL

R Personal freedoms and individual self-determination became major planks in the Democratic platform around this time and replaced most of the old collectivist planks in the platform with the exception of health care and education, which were too useful to the left to discard as allies. If I had to pick the one grandest change in the left from the pre-Cold War era to this one, that's probably it rtight there.