Ancient Celtic-speaking peoples did, actually, have a whole variety of writing systems for their languages that they directly borrowed with one exception, to their neighbors.

The first scripts used to write down Celtic languages were variants over the Old Italic script used by Etruscans and thus called Gallo-Etruscan (or Lugano Alphabet). Earlier evidence are found in Lombardia, at the core of an archeological horizon known as the Golasecca Culture, for the VIth or maybe the VIIth centuries BCE.It is not sure how related the Celtic language recorded in epigraphs, named Lepontic ,was to other continental Celtic languages but is is often seen as both a distinct branch from other known mainland languages if close to Transalpine Gaulish, but as well a possible archaic variant of the Celtic language spoken in northern Italy from the IVth century BCE onward known as Cisalpine Gaulish.A possibly related language, we have too few elements to really categorize, would be Noric attested by two fragmentary inscription in a different variant of Old Italic script.

Hispano-Celtic languages, such as Celtiberian and possibly Tartessian (altough this identification in particular remains disputed) used several variants of Iberian script, originally adapted to Iberic languages from either or both Greek and Phonenician alphabets. At the exception of Tartessian, Celtiberian written evidence appears tardily around the IInd century BCE significantly delayed compared to the adoption of writing by south-eastern and eastern Iberian-speaking peoples.Other languages considered as possibly Celtic and more directly related to Celtiberian are Gallaceian and Lusitanian, written using Latin script in the turn of the millennium.

Although two fragmentary inscription in Gallo-Etruscan were found beyond the Alps, probably indigenously made, the first main script used in Transalpine Gaul to write Gaulish was the Gallo-Greek script, based on Ionian Greek alphabets, used from the IInd to Ist centuries BCE in southern-eastern Gaul, especially around the delta of the Rhone. It was eventually replaced by the use of Latin script called Gallo-Latin between the Ist century BCE and the IIIrd century CE.Across the Channel, a seemingly close British language is poorly attested in written forms using Latin script, essentially fragmentary or mixed with Vulgar Latin.

Eventually primitive and old Irish, possibly together with Pictish, used an indigenous-made script called Ogham first used in ostentatious epigraphs since the IVth century CE, maybe at the imitation of Latin script an as a display of prestige finding its origin either in southern Ireland or even in the Irish-speaking communities present in post-imperial western Britain. The competition brought by the introduction of Latin script by missionaries and the christianization of Ireland competed with the use of Ogham which slowly declined retaining its cultural prestige.

(Partly taking from this earlier answer at this point)

At first glance, we can say that yes, ancient Celtic-speaking peoples did used writing in various forms. But we can point to some common features in spite of the cultural, linguistic and societal differences that existed between peoples separated by centuries and long distances.

1. At the exception of Ogham, which might have been made at the imitation of Latin, all of these alphabets were borrowed from neighboring peoples.
2. Writing was adopted significantly tardily compared to these, at the exception of northern Italy, most of them appeared no earlier than the late IIIrd century BCE, while the competition with Latin soon made writing them obsolete, some of them were only written down after the Roman conquests.
3. At the exception of Ogham variants, writing was mostly reserved for ostentatious (calendar, coinage, funeral or votive headstones, etc.) or for practical purposes (advertisement, censuses ,contracts, property marks, etc.). "High literacy", which is the practice to write down and read important literary production such as epics, poems, religious or philosophical beliefs, history or folklore was unknown to ancient Celtic-speaking people.