There's no specific mention of what the new religion is (indeed I'm not certain there is any specific mention of it at all), and the show is notably devoid of specific Christian iconography usually present in Arthurian legend. The "Cup of Life" is quite obviously analogous to the Holy Grail, but in Merlin it is a Pagan object, not a Christian one.
Similarly, we see no clerics (other than those of the Old Religion). Services that would, even today, often had a religious component (weddings, coronations) are entirely secular, usually presided over by Geoffrey of Monmouth. However, Arthur does make mention of continued use of some Pagan feasts (Beltaine, Samhain), implying that non-magical customs from the time of the Old Religion are still permitted/observed.
Out of universe: Was "new religion" completely unmentioned on purpose?
Almost certainly. Making Uther explicitly (even if not devoutly) Christian would highlight his oppression of the druids in ways that would have presented difficulties for the show: it would likely have been a turn-off for many viewers, for example; it may also have been difficult to reconcile the often humorous tone of the show with a serious exploration of a historical clash of two real-world religions.
However, it should be noted that the writers were not nearly so circumspect on the other side of the equation: they frequently use (and indeed misuse) Pagan terms and symbols.
The Arthurian legends were set during the late 5th to early 6th
century. What was the predominant religion in Camelot during those
Camelot is not a known historical place, but 5th-6th century Britain was under going changes on many fronts -- invasions, new kingdoms, plagues, and yes, conversation to Christianity from pre-Christian Pagan faiths (about which much is unknown).