/sɛlt/ are considered acceptable pronunciations of the noun Celt and similarly of the adjective Celtic. Is there a reason for the different pronunciations? Which is the more common? Is each variant perhaps particular to a geography? Scottish vs. Irish, maybe?
According to OED 1, Celt is first recorded in English in 1607, probably Anglicized from Latin. At that time it designated the peoples whom the ancients called Celtae (Latin) or Keltoi (Greek): the Gauls and those of Spain and Northern Italy “believed to be of the same language and race”. Celtic first appears in Blount’s Glossography as a French word, Celtique, defined as “pertaining to the people of Gaul”. The modern sense arose in the 18th Century; it
began in French, and in reference to the language and people of Brittany, as the presumed representatives of the ancient Gauls: with the recognition of linguistic affinities it was extended to the Cornish and Welsh, and so to the Irish, Manx, and Scottish Gaelic.
(That recognition, it might be remarked, was in great part due to the labours of the undeservedly neglected Welsh naturalist, antiquarian, and Keeper of the Ashmolean Museum Edward Lhuyd (his reformed spelling of the name “Lloyd”). Lhuyd’s Archaeologia Britannica, 1707, laid the foundations of Celtic linguistics and contributed to his countryman Sir William Jones’ epochal recognition, three generations later, of the kinship of what we now call the Indo-European languages.)
There is no reason to think that the pronunciation at that time was anything other than /ˈsɛlts/, as in both French and contemporary Englishing of Latin. Scholarship had generally recognized at least as early as Leibniz (Collectanea Etymologica, 1717) that the name should “properly” be pronounced with a /k/ (the characterization suggests that the usual pronunciation was with /s/), and spellings with /K/ begin showing up among speculative antiquarians at the end of the 18th Century. These spellings gain a modest currency in the 19th century, but (according to NGrams) never offer a serious challenge to spellings with /C/.
We may presume, I think, that in the 19th Century the pronunciation with /s/ likewise dominated that with /k/. OED 1 (the Cast-Clivy fascicle was published in November 1889) gives this pronunciation first, followed by “Also Kelt (kelt).”
Since the 19th Century, spellings with /K/ have almost disappeared; but my experience is that pronunciations with /k/ have gained ground over the last 50 years.
The pronunciation with /k/ has existed as a minority variant alongside the pronunciation with /s/ for over 200 years. As The Celtic Wiki hints, it is mere linguistic politics to stigmatize either as “ignorant”.
Following the rules of English pronunciation, the sound of "C" is based on the following vowel e.g. celery, city, cat, cot, cut.