'Norse' people usually signifies the peoples (plural) who spoke the Old Norse language; Danes, Norwegians, Geats, Gotlanders, Swedes, etc in the period when that language was use; or in other words as a byword for Medieval Scandinavians roughly 700-1200 or so, preferably in contexts where it makes sense to do so. That is, the common features of these cultures.

One common feature is the runic writing system. But it was not developed by the 'Norse'; as it was used for centuries before it makes sense to speak of the Norse and by all Germanic peoples. It may have been developed around the first or second century CE, and the first inscriptions are from about the third or fourth centuries. Those early inscriptions are in proto-Germanic, and in Gothic (an East-Germanic language). So it was in use by East, West and North Germanic peoples. That also includes the Anglo-Saxons, who'd bring the writing system with them to England and continue to develop their distinct version for some centuries before the Latin alphabet won out. So runes are by no means unique to or developed by the Norse, they just remained in use in Scandinavia longer. (and continued to change as long as they were in use; Medieval runes are different from Viking Age ones, are different from proto-Germanic (Elder Futhark) ones).

The runic writing system was not an independent creation but inspired by the Latin or possibly another Italic alphabet. First because a number of letters are obviously the same ('t', 'b', 'f', 'r' are essentially identical to their Latin counterparts) and others have similarities. It also comes into being in a period known as the Roman Iron Age in nothern Europe, because it's a period of strong Roman influence on Germanic societies. However there's enough difference for it not to be thought a direct copy of the alphabet. For instance, the ordering of letters is different and that's something that's otherwise highly conserved (from Latin A-B, to Greek Alpha-Beta, to Hebrew Aleph-Beth, etc)

Now when it comes to the Celts, that's an even bigger ethnolinguistic group that stretched from Romania across the continent to Ireland. But assuming here you only mean the Celts of the British Isles, and during the time period of the first millennium AD in which runic script came into use, then the answer is simply that they did develop their own system of writing: Ogham script. Ogham inscriptions have been found in pretty much the whole Insular Celtic area largely outside the areas settled by the Romans and later Anglo-Saxons; Ireland, Scotland, Wales, Cornwall, Isle of Man.

The earliest Ogham inscriptions date from about the 4th century CE, making it possibly somewhat later than Runic script but of the period, when the Romans had expanded into Britannia and Germania Inferior and were in proximity with Ireland as well as north Germany and Scandinavia. It's also in the first centuries BCE and CE that the first references to Ireland and Scandinavia show up in the writings of Roman geographers like Strabo and Tacitus.

Compared to Runic, Ogham is much more distinct in appearance from the Latin alphabet. But it is nevertheless though to have been inspired by it; it would for certain have been the writing system they'd have come in contact with. But its exact origins are entirely conjecture at this point.

Completely independent inventions of writing are few and far-between historically; There's the Sumerians, the Mayans in America, probably the Chinese and maybe Egyptian hieroglyphs. Everything else we know of is thought to be based on or inspired by contact with an existing writing system.

Just to expand on the above answer, the idea Celtic languages were never put down in writing before the Roman conquest is a common misconception. Localised forms of the Etruscan alphabet were used in Gaul and around the Alps. Gaulish would also be written in Greek around the area of Massalia, modern-day Marseilles, a Greek colony. Celtiberian was written in the Iberian script, a Phoenician-based syllabary it shared with other non-Indo-European languages. There is also the case for the Southwest Script but it remains highly controversial. A more detailed but not too extensive outline of the different scripts used to represent Celtic languages can be found in John T Koch’s Celtic Culture: a Historical Encyclopedia (2006 pp 1594-1600).

This said, we cannot ignore the use of the Latin alphabet, given that Romanisation did not just happen overnight, and many conquered Celtic peoples from Gaul to Britain and Iberia started using it to represent their own vernaculars, sometimes giving us the only attestations of their languages before they were displaced by something else.

When it comes to Ireland, the Roman conquest of Britain definitely affected their neighbors on the next island. The archaeological record in the first half of the Iron Age (ca 500 BC to 0AD) shows a generalised decrease in agricultural output, monumental construction and virtually no evidence for long-term settlements.

This changes at the turn of the first millennium AD with the Roman conquest of Britain. At this time start seeing the appearance of exotica in the material record such as the skull of a Barbary monkey in Navan Fort, as well as the monumental constructions or modifications of the ‘royal’ complexes of Ireland. Archaeologist Conor Newman argues this was led by a class of ‘nouveaux riches’ who probably rose to power through raiding and trading with their Romano-British neighbours. Since Irish society was coinless at the time, richness was expressed through cattle and slaves, something that shows up poorly in the archaeological record.

As a side note, the word Goidel, which gives us the modern Gael, is itself a borrowing from an Old Welsh word for ‘raider’. In any case, the Roman world definitely left an impact in Gaelic Ireland, albeit a less visible one than what we have for its continental counterparts.

Ogham scripts are a Christian, Medieval phenomenon. We find their parallels in commemorative stelae that were common in what would later become Wales and England at this time; and which may have served as the source of inspiration for the ogham stones seen in Ireland and later in Scotland and Wales.

Sources:

Cunliffe, B. 1997. The ancient celts. Oxford University Press.

Halpin, A. and Newman, C. 2006. Ireland: an Oxford archaeological guide to sites from earliest times to AD 1600. Oxford Archaeological Guides.

Koch, J.T. 2006. Celtic Culture: a Historical Encyclopedia. Santa Barbara: ABC-CLIO

Newman, C. 1998. ‘Reflections on the making of a “royal site” in early Ireland’. World Archaeology, 30(1), pp.127-141.

Rankin, D. 2002. Celts and the classical world. London: Routledge.

Hey there,

Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.

If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!