If Ajani's Pridemate is dealt lethal damage at the same time that you gain life, it won't receive a counter from its ability in time to save it.

In a Two-Headed Giant game, life gained by your teammate won't cause Ajani's Pridemate's ability to trigger, even though it caused your team's life total to increase.

Each creature with lifelink dealing combat damage causes a separate life-gaining event. For example, if two creatures you control with lifelink deal combat damage at the same time, Ajani's Pridemate's ability will trigger twice. However, if a single creature you control with lifelink deals combat damage to multiple creatures, players, planeswalkers, and/or battles at the same time (perhaps because it has trample or was blocked by more than one creature), the ability will trigger only once.

The ability of Ajani's Pridemate triggers just once for each life-gaining event, whether it's 1 life from Dazzling Angel or 4 life from Apothecary Stomper.

If you gain an amount of life "for each" of something or "equal to the number" of something, that life is gained as one event and Ajani's Pridemate's ability triggers only once.