It doesn’t matter who controls Hive Mind and it barely matters who cast Glorious End. What matters more is whose turn it is.

When a player casts Glorious End here, each other player will put a copy of it on the stack as Hive Mind’s trigger resolves. These copies go on the stack in APNAP order. So the player whose turn it is, if they didn’t cast Glorious End, will put their copy of Glorious End on the stack, and then the next player in turn order will put their copy on the stack above the previous player’s copy, etc.

The net result here is that the player who didn’t cast Glorious End and most recently finished their turn (i.e. last in APNAP order) will have their copy of Glorious End resolve first. This ends the turn and exiles all the other copies and the original spell, which won’t resolve. Only that player will have a “lose the game” trigger occur in their next end step.

For example, A, B, C, and D (in that turn order) are playing a game. A controls Hive Mind. B casts Glorious End on D’s turn. When Hive Mind’s trigger resolves, D puts a copy of Glorious End on the stack, then A’s puts their copy above D’s, and C puts their copy above A’s. C’s copy will resolve and exile all other objects on the stack. Only C will have a “lose the game” trigger on their next end step.