No, not really.
The populations of the colonies would show the imprint of the colonization process. In some settings (and some real world examples) colonization is driven by some religious or cultural minority searching for a home where it forms a majority. Such colonies would be fairly uniform in religion or culture.
Unfortunately in your setting the colonization was driven by an effort to evacuate survivors so that they do not die. The priority would be entirely on survival with cultural aspects being relegated to the "next generation, if it exists, can think about that" category.
Depending on the specifics this might not erase large existing cultural groups. For example if evacuation is organized by national governments, you'd reasonably have a Chinese colony, an Indian colony and so on. In addition to national pride and governments wishing to retain power it would genuinely be more effective to avoid jurisdiction issues by mirroring the political structure here there.
But you specified relatively small number of survivors. I do not think it plausible that most national governments would have survived intact. At best you could have a situation where survival was based on whether your government had the resources to get you off the planet. In which case we'd have only the cultures of the most powerful nations left.
More probably the evacuation would have been a global effort. The survival of the species is at stake so you want to evacuate everyone and everything you can with hard priority on maximizing the odds the humanity survives on its refuge. There would probably be single evacuation authority and single colonial administration using human resources based on utility value with culture, religion and nationality seen as irrelevant. Next generation, if one exists, will think about all that.
So there would IMHO be a single nation of survivors with fairly uniform culture and values formed by the struggle to survive. While there would be multiple colonies scattered over large area and they'd have their own local administration, valuable human resources would be moved as needed.
It might be useful to contrast this to the US. It would have similar melting pot character as US had but instead of large groups of people moving over and settling in it would have traumatized survivors settling and working where they are needed. There would be problems retaining families and major religions intact, cultures or nations would be a bridge too far.
But the next generation does exist
So how would the new generation seek to address this?
First thing to understand is that they all grew with the new unitary culture of survivors and strongly identify with it. First generation might have parents who were of same Earth nationality and culture. After that, people would just be aware of where the grandparents came from. They'd probably be interested in finding their roots and learning about the culture of their grandparents. All of their grandparents.
So instead of colonies dedicated to single culture, I think more reasonable model would be the way Japanese handle religion. Mix and match. One ritual or celebration according to one model, another according to another. After all you would want not to lose the old traditions.
I mean you would not want to lose the tradition of "suojalkapallo" or "eukonkanto" just because nobody remembered to invite the Finns. Or the Maori culture just because nobody could figure out where New Zealand actually is. And is it even a real place if it is not on the map?
So everybody would happily borrow from all cultures and do their part to remember all of the past.