Why is Wales not represented on the union flag flag?

Because at the time of the formation of the United Kingdom Wales was considered to be a Dominion of the English Crown (as it had been since 1535). Note that although the British monarch's eldest son bears the title Prince of Wales, he plays no role in the governance of the country (which it is today held to be) so that Wales is not a principality.

The Welsh flag is white on its upper half and green on its lower half with a red dragon passant. This flag can be flown alongside the Union Flag, but not above it, in Wales.