My Undergraduate dissertation and Master's thesis both concentrated on swords from the Irish Iron Age (c.700 BC- c. AD 400) and I can say with confidence that two-handed swords were not a feature of the Irish Iron Age (or Early medieval period). In fact, the defining feature of Irish swords from this period is that they are, in comparison to British and continental European swords from the same period, exceptionally small in terms of blade length; the longest complete example of a La Téne sword in Ireland is about 55cm (including handle) and there is one example with a blade only about 25cm long. Later Iron Age swords based on Roman weapons max out at around 62cm in overall length and Early medieval swords are a little longer but not by much.

The two handed sword, in Europe at least, is a feature of the later medieval period (I believe it's in the late 1300's that what we now call a longsword really comes into existence) where armour technology had evolved to the point that a shield was not necessary for protection when wearing full harness and you could use a two handed weapon at close quarters and still enjoy a good level of protection.

While armour certainly existed in Iron Age Europe (though we have no evidence of it in Ireland) full plate was still a long way off and the longsword with it.