1️⃣ Combat Damage Happens Simultaneously
When combat damage is dealt:

The lifelink creature deals damage.
It receives damage at the same time.
If it takes lethal damage (damage ≥ its toughness), it now has lethal damage marked on it.

2️⃣ Lifelink Gains Life Immediately
Lifelink doesn’t use the stack.
You gain the life at the same time the damage is dealt.

So yes — you gain life right away.

3️⃣ State-Based Actions Are Checked
After damage is dealt and lifelink life is gained, the game checks state-based actions.

At this point:

If the creature has lethal damage marked on it
And nothing is preventing it from dying
→ It is destroyed.
Your life total going up (even from lifelink triggering something like Leyline of Hope) does not remove the lethal damage already marked on the creature.

4️⃣ About Leyline of Hope
Leyline of Hope triggers when you gain life. That trigger:

Goes on the stack after damage is dealt.
It does not resolve before state-based actions are checked.
So even if Leyline of Hope would later put counters on the creature or otherwise help it, the creature will already be dead before that happens.

Lifelink gaining you life doesn’t save the creature from lethal damage, and Leyline of Hope’s trigger resolves too late to prevent it.