Both creatures' abilities will actually trigger; it doesn’t matter that one dies.
The key is that all damage is done simultaneously, and with Lifelink, the life gain happens simultaneous to damage dealt. So at the moment you gain life, both creatures are still alive, even though one of them has lethal damage now.
603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
The creature with lethal damage doesn’t die completely immediately; it dies the next time state-based actions are checked, which is before anyone would get priority. When state-based actions are checked, the abilities that were triggered then go on the stack.
704.3. Whenever a player would get priority (see rule 116, “Timing and Priority”), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event. If any state-based actions are performed as a result of a check, the check is repeated; otherwise all triggered abilities that are waiting to be put on the stack are put on the stack, then the check is repeated.
704.5g If a creature has toughness greater than 0, and the total damage marked on it is greater than or equal to its toughness, that creature has been dealt lethal damage and is destroyed. Regeneration can replace this event.
The creature will be dead by the time they go on the stack, but that doesn’t matter.
112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won’t affect the ability.
