Both Puddings Are Slowed
In this case, both puddings should be slowed. This isn't specific to being slowed, it should be true of any condition or spell effect.
Spells and conditions have definite points at which they terminate. A spell doesn't end unless a certain condition is met (the duration is met, counterspell, etc.) - and the Split ability isn't one of them.
I'm going to walk through this inductively, since there isn't a single rule to reference:
Both puddings are either afflicted with the spell or condition, or neither is.
The text of the black pudding's ability says (emphasis mine):
Split (Ex) Slashing and piercing weapons deal no damage to a black pudding. Instead, the creature splits into two identical puddings, each with half of the original's current hit points (round down). A pudding with 10 hit points or less cannot be further split and dies if reduced to 0 hit points.
If one of the new puddings is slowed, but the other isn't, than they aren't identical.
Both puddings must be slowed, since the Slow spell hasn't ended yet. Two possibilities remain: either both puddings are slowed, or neither is.
Slow has a duration of 1 round per level. Without some other effect that removes slow (such as Haste), it stays until the duration is met. The black pudding's Split ability does not explicitly remove any magical effects or conditions.
This interpretation follows from the rules text, but relies on the non-existence of a clarifying rule. If someone does find a specific rule that governs this, it would be authoritative.