Okay, I looked at some right-wing articles to see what they are claiming as woke policies from Biden:
An executive order in June 2021 calling for federal agencies to prioritize diversity, equity, and inclusion in recruits and when considering promotions.
Allocating Covid recovery funds on the basis of race and gender:
(Spoiler: it was not constitutional)
Opposing the lawsuit alleging that race-based affirmative action at Harvard and UNC was unconstitutional.
Reinstating Obama-era rules that remove due process for anyone accused on sexual assault or harassment on university campuses:
There are a few more similar but I’m getting tired.
Sure. And many of them were planning to turn London into Singapore-on-Thames or something, which was not at all what most Brexit voters wanted. But despite the hypocrisy, I think there was something to the charge. As far as my research could tell, the EU really did economically benefit skilled/higher social class workers far more than low skilled/lower social class ones, while the later bore the brunt of the social and structural problems from mass migration as usual. We get to pay less when we need a plumber or building work, they get lower wages and fewer opportunities.
Illegal immigration in the US is similar: employers get cheap labour, poorer people see lower wages and more competition (probably the reason some Latino citizens are not big fans).
Pretty much everyone I have seen agrees that the border was a major reason Harris lost the election. Whatever they say in surveys, Americans are clearly unhappy with the current results. 
Not in itself. This is woke:
Accusing people of xenophobia or bigotry for wanting to be able to decide who moves to their country and in what numbers is very woke.