I can answer the Irish dimension of this question, but I'm afraid my knowledge of Britain in the same period isn't enough to answer satisfactorily.

Without going into the "is it correct to call the Late Iron Age peoples of Britain and Ireland Celtic?" debate, I assume you're asking about roughly 400 BC to AD 400, which is from the beginning of the 'developed' Irish Iron Age when we see La Téne type weapons, personal ornament and other items appearing in Ireland, to the end of the Iron Age in AD 400 with the beginning of the (Christian) Early Medieval period.

The first thing to understand is (I find) that there is often a disconnect between broad archaeological studies of weaponry in La Téne/Celtic Europe and (older) studies done within Ireland that touch on the subject. For example, in Pleiner's 1993 The Celtic Sword he supposes Irish warfare into the Early medieval period remained unaffected by Roman influence and could be assumed (with no supporting evidence) that the mode of battle that the Greeks and Romans described Keltoi and Galli employing (a headlong charge with long slashing swords) could be transplanted to Iron Age Ireland. On the other hand we get Irish studies supposing on the evidence that Irish La Téne swords are almost comically short (30cm long in some cases with a maximum length of about 55cm, including the grip) that Irish Iron Age warfare was either non-existent or a "Ritual farce" (Scott. 1990) and weaponry was made for show.

Clearly the idea that the Irish fought in a similar way to Continental people or even British people is probably wrong, for one, there's the swords; compared to the Irish ones as I described above, Continental La Téne swords start at about 60cm long at their shortest (Short continental La Téne swords are usually a specific type called the anthropoid-hilted swords) and by the Late La Téne period they are topping out at over 1 metre long, with many types clearly being designed for cutting prowess over piercing or stabbing: there are types which lack a point altogether and just come to a sudden stop Like this example from Denmark. Similarly in Britain we see the progression over the La Téne period to swords about as long as the Continental types.

In Ireland we don't have that progression, we have swords with blade cross-sections that would seem to be Late La Téne types but are still really short. Unfortunately we have only one dated example of an Irish La Téne sword - from Dún Áilinne in Kildare, but the dating at that site is thrown off by a problem the Belfast radiocarbon lab were having at the time. All La Téne swords in Ireland come from wet places like rivers, lakes or bogs (probably deposited in what was a bog-pool that became overgrown by the bog) except the Dún Áilnne sword, which comes from a foundational phase at what is interpreted as a ritual site.

However, with spears we have a little more in common with the continent: no Irish La Téne spearhead would be out of place in a British or continental assembly, again, all are from wet contexts, bar one from Dún Áilinne.

We have no armour dating to the Iron Age and one shield tentatively dated to the Iron Age on the basis that it doesn't resemble any other period shields: it is rectangular and quite small (57cm X 35cm) compared to Continental and British shields from the period. It is constructed of wood covered by leather.

Before I finally get around to actually answering your question, I just want to say that unlike on the Continent and in Britain we do not have the phenomenon of the weapon burial (bar one on Lambay island but it was found by workmen in the early 20th century and the sword was not treated carefully enough for it to survive), which gives you lots of date-able organic material in conjunction with the sword. This is both really interesting and another point of departure Ireland has with the rest of La Téne Europe, and rather annoying, because it means most Irish La Téne weapon finds are older finds from river dredging works, peatcutting, and lake drainage for land improvement from the late 1800's to the 1950's. This really impacts the quality of preservation (one La Téne sword was given to the museum by a guy who had been using it as a tyre lever for his car, and another was discovered in "the thatch of a disused cottage" - which is super interesting for someone studying the reuse of archaeological artefacts in later religious/magical/apotropaic contexts but not so fun for us trying to get contextual archaeological information) and it also means that most were removed from their archaeological context and no archaeological work was done in the surrounding area with some honourable exceptions.

Okay, so, what would Irish Iron Age warfare have looked like? Probably small-scale compared to Britain and the continent: Ireland underwent a population decline at the end of the Bronze Age which only seems to have started to reverse around 300 BC. We see from pollen records that arable agriculture and cleared land species appear to have declined in favour of enclosed woodland species, showing that woodland had started to encroach on areas previously kept open by humans for arable and pastoral agriculture. When population picks up again, it is still relatively dispersed, without the trend in Southern Britain for hillforts or the continental Oppida. While you do get Iron Age re-use of Bronze Age hillforts in Ireland, this doesn't appear to be defensive or even permanent settlement in nature. From AD 300 you do start to see enclosure of settlements, usually just one or two buildings, which turns into the ringfort phenomenon more associated with the Early Medieval period.

In a dispersed population that probably practiced some degree of mobility (not nomadism as was sometimes mooted, but seasonal transhumance for farmers) like this warfare was probably mostly directed at raiding for cattle. The lack of long swords would indicate to me that cavalry was not widely used (no Irish Iron Age sword is long enough for use on horseback). While spears would have sufficed, spears do break and there is no reason not to have a backup weapon. Warfare, when it took place, was probably between small groups, on foot, with one group attempting to take the movable wealth (cattle) of another group and the others attempting to defend it. We don't have fortified central places where sieges may have taken place, and lack of armour and large shields would indicate that heavy infantry were not a feature of Iron Age warfare either.

I hope that answers your question.

Sources:

Dowling, G., 2014. Landscape and settlement in late Iron Age Ireland: some emerging trends. Late Iron Age and ‘Roman’ Ireland, pp.151-174.

Pleiner, R. 1993 The Celtic Sword. Oxford. Oxford University Press

Raftery, B. 1983. A catalogue of Irish Iron Age antiquities. Marburg. Wasmuth

Scott, B. 1990. Early Irish Ironworking. Ulster Museum Press.