St Patricks Cross

On the Union Jack you will see the Red Cross of St George (edged in white), the white diagonal cross of st Andrew and the red diagonal cross of st Patrick. St Patrick's cross does not appear on either the Northern Irish flag nor the Eire flag, interestingly enough. The Union Jack heralds from a time when both Britain and Ireland were united and remained the same even when most of Ireland became independent in 1921.

You are quite correct about Belize... Britain simply has a small military presence there nowadays (presumably due to Guatemalan disputes over territory and maybe due to the Queen still being the head of state?) and it is no longer a dependent.