As pointed out before, Celtic paganism was moribund if at all existing during this period, and was largely replaced by the somewhat syncretic Celtic Christianity.
For reference, Semitic paganism would be more sensible to add due to "mentions" of continued practice in Yemen into the 19th century, mention among certain nomads by Persians in the 16th century, and the known majority population of Harran being pagan in the CM start date. Hellenism has the Mani peninsula, which currently lacks any holding representative of it (no barony, no church, and no city), and was still pagan in the CM start date.
Especially the indigenous Heathen religions (Celtic, Germanic/Asatru and Slavic/Rodnovery). I can personally verify that to we Asatruar, religion and culture are indistinguishable.
I cannot 100% verify the invalidity of this, not being an Asatruar myself, but I'm not entirely certain this is the case, given the multicultural Germanic presence, as well as certain aspects having little secular value. This is, however, not to discount the large impact of traditional beliefs on such cultures, which have largely shaped them and still do today, regardless of the Abrahamic dominance in said regions.
I'd say Judaism is perhaps the best example. Almost all of the holidays and festivals within it are representative of historical (or believed historical) events. Passover, for instance, celebrates the return to Canaan, rather than the divine occurrence associated with it. Hanukkah deals with the victory over the occupants and the supposed miracle of long-lasting oil, not of divine interference in said war. Purim celebrates an escape from potential extinction at the hands of a malicious Persian.
Meanwhile, all of the practices likewise hold secular origin representative of preislamic, especially Canaanite (Phoenicians and Hebrews), with clear cognates in Akkadian and Egyptian practice. The Kosher diet, for instance, is about maintaining health. Many of the things forbidden by Kosher are so because, in ancient days, they led to potentially fatal sickness and parasites. Many of the rituals in Judaism are about cleanliness and sanitation, from when you can pop a baby into a lady to circumcision, the whole lot is about staying clean and healthy. This is where Egyptian cognates are the highest, as the Egyptians likewise had a very strong cultural emphasis on cleanliness (and, when one lives in the desert, it is to be expected. Sand everywhere, ugh.) Other rituals, practiced in the past and today, represent ancient concepts like tribal taxation, tribal dominance, and other such political mechanisms.
Is there a religious aspect to Judaism? Of course. There's multiple texts written about the religious aspect, although it is hardly a static aspect. The Hebrews originally had a religion identical to the Phoenicians, and the two languages were near identical (even today, Hebrew closely mimics the long-dead Phoenician). Later, they began to worship the king of the gods, El, while recognizing the existence and powers of others. This henotheism would continue well into the Roman age. Some time in the early iron age, the name "Yahweh" began to appear. It has its origins as an Arabian (specifically from Midian, later known as Nabataea) deity associated with El. Due to the frequent trade and similarities between the two deities, both "El" and "Yahweh" were used by the Hebrews to refer to the same deity. It was not until the Middle Ages that Judaism reached a form of true monotheism, and even then it was not universal.
My general point is that the rituals and festivals in Judaism can be seen outside of it. The ties to historical events and secular culture, with clear cross-cultural and cross-religious cognates, demonstrates within Judaism a very close tie to religion and culture. Although things like Passover, Hanukkah, and Purim are seen as aspects of Judaism (indeed, almost never observed outside of it), they were spread across all Hebrews, pagan or monotheistic, Hellenized or Jewish, the nomad and the cities, the peasant and the king, universally. The festivals of Judaism transcended the religion and stuck within the secular culture, as their festivals had little, if anything, to do with their religion. It is only after the diaspora that such practices stopped, as the Hebrews warred with Rome multiple times and ultimately lost, losing their homeland and having many of their people killed. The Hebrews became almost exclusively Jewish at this point, as the ones left behind abandoned their culture in the 5th century in favor of the Greeks, becoming Christian Palestinians, and then to the Arabs just a couple centuries later.
And so the diaspora essentially ceased the existence of non-Jewish Hebrews. If they converted, they usually adopted the local culture to a much higher degree than non-converts as well. However, the modern state of Israel is allowing for a return of multireligious Hebrews. Atheists, Buddhists, Pagans, Jews, Christians, Muslims: all exist now among the Hebrew population, and given that they exist as ethnic Hebrews, they almost always follow the celebrations of Judaism.
Here we see multiple faiths taking up Jewish religious practice. Judaism took the way of life of preislamic Canaanites and gave it religious justification, rather than incorporating religious practices into the culture of the locals. This, my friend, is the mark of religion and culture becoming near indistinguishable.
