dnd 5th – Do the damage of the absorbed items apply to your next attack or should they be on your next turn?

The first sentence of Absorb the elements has no qualifier

The language at the head of the spell simply indicates that the spell is

store it for your next melee attack

This seems to work well in your scenario, but it is also in direct contradiction with the third line of the spell.

But then it does …

The third line of the spell adds a qualifier in which additional damage is triggered at the beginning of your next turn if you perform a melee attack.

I have bolded the relevant qualifier in the quote below:

Also, the first time you hit with a melee attack on your next turn, the target takes 1d6 more damage from the trigger type and the spell ends.

In this reading with your scenario, you could not use it in the same trick as you used your reaction to launch it.

Until the DM

My not demonstrable intuition is that the assumption is that you receive your damage when it's not your turn and that you will not usually get an attack before your turn. In your scenario, you trigger that damage yourself by jumping into the bonfire.

Given that, I do not think it would be unreasonable for a DM to authorize its use in the same round. You have chosen to expose yourself to damage and burn a resource to inflict more damage. Just be aware that if you have a concentration spell in progress, you will also need to save a backup for the fire damage you do.