To an Irish born father and mother so he could be considered a dual citizen. In any event his family returned to Ireland when he was very young. He grew up there, was educated there, and spent his entire adult life living and working in England before fleeing to Nazi Germany about a month before the war began. His connections to Britain were infinitely stronger than Brown's connections to Germany, and his execution for treason by the British makes perfect sense.
There was also some legal technicality about Joyce having taken out a British passport at some point and as a result having assumed an obligation of allegiance to the Crown.
He was therefore charged for treason only for his acts during a relatively brief period of time during which the legal technicalities were in effect.