Note that while you posit the smaller puddings using their reaction, if should they get one, for an opportunity attack, there is a more likely use for their reaction: a second split! If the new puddings receive an appropriate attack before their initiative, they would be able to split again, provided they come into being with an unused reaction. If the mother pudding starts as size Huge (such as the ones in Tales of the Yawning Portal p.185), this could even happen a third time in one round.
Do you want them to have a reaction?
If you would like them to have a reaction, they do. As ADdV says, "they still have their reaction, as these new slimes have not yet used a reaction in their entire existence." You are effectively adding new creatures to the combat.
If you would not like them to have a reaction, they still do - but they can't use it because they are surprised.
I was trying to find some information on their place in the initiative order, hoping that it can help me out, but no luck with that either.
The rules for combat assume that all participants are there at the start, but this is obviously not always the case, resulting in numerous questions about how a DM should handle late-joiners (see list at end).
While there is no RAW for joining a combat already in progress, site consensus seems to be that new participants in the combat should roll initiative as they come in and then join the initiative order. However, I would take this a step further and advocate that new additions to the combat should be onboarded by having them run through of the entire Combat Step by Step (PHB p. 189) until they are at the same place as the other creatures already in combat:
- Determine surprise.
- Establish positions.
- Roll initiative.
- Take turns.
- Begin the next round.
1. Determine surprise. Normally someone joining a combat would be aware that it was going on - they saw the action, heard the noise, and probably arrived with the intention of joining. But consider someone turning a corner and coming on a combat already underway but within the effects of a silence spell. Or a wizard teleporting to her 'safe' laboratory and finding her guardians already engaged with attackers. In this case surprise would be reasonable. Likewise, if you as a DM do not want the new daughter puddings to be able to use reactions on the turn that their mother pudding split, you could easily say that the adjustment to a new body, loss of half their chemical memory, and different sensory input has rendered them surprised for the first round of their existence.
2. Establish positions. You likely did this intuitively anyway, removing the old pudding from the map and putting the new ones in the same or adjacent spaces.
3. Roll initiative. As above, the new puddings are joining the combat - you could choose to keep them on the old pudding's initiative for convenience, or roll a new one for both, or roll a new one for each.
Now that you have finished adding the new puddings to the combat, consider their options. If a foe is leaving their reach without disengaging, they could use their reaction to attack since they have not previously used their reaction. But if they are surprised, they
can't move or take an action on [their] first turn of the combat, and [they] can't take a reaction until that turn ends.
So, while they have reactions, they can't yet use them.
How do I handle initiative when a new force joins a combat that's already in progress?
How do you add extra waves of NPCs to an ongoing fight? [duplicate of the previous]
An initial stealthy/surprise attack with subsequent adventurers entering combat afterwards?
Can you choose to delay rolling your initiative?
What happens when you are hidden and a new enemy joins initiative?
How does initiative work when teleporting into combat?