Is there a cultural/historic reason why Wales has a more complex looking flag than England, Scotland or Northern Ireland?

Different reasons. 

So to start with England, the quite heraldically-complex Royal Standard (shown below) was gradually de-emphasized, first because the Stuarts wanted to emphasize the personal union between England and Scotland, and then after the Civil War and the Glorious Revolution of 1688 a desire to de-emphasize the royal family in favor of the nation-state, and then after the Act of Union of 1707 the need to emphasize this newfangled concept of Britishness.  

In Scotland, the Royal Banner which you can see on the top right quadrant there was de-emphasized by the Scottish Parliament in 1672 in favor of the St. Andrew’s Cross/the Saltire, to emphasize the distinction between the King of Scots and the kingdom of Scotland (given the fact that the latter had warred quite successfully against the former during the Bishops’ Wars  and the “Wars of the Three Kingdoms”). 

And in Northern Ireland, the “Ulster Banner” was discontinued after the imposition of direct rule in 1972, and after the end of direct rule in 1998 its use by government entities was avoided because of the flag’s association with unionist/loyalist politics, although it’s still used in international sports.