The rule for tiny creatures does not specify only "naturally" tiny creatures. It is a blanket statement -- if you are tiny or smaller, you use your dexterity modifier. Other rules that apply to tiny creatures apply to creatures made tiny by reduce person (bonus to AC, penalty to STR, etc), and I see no reason to think that this particular rule is any different. The wording for tiny creatures who want to make a climb check is clear, in that it applies to any tiny creature.
Any creature of Tiny or smaller size should use its Dex modifier
instead of its Str modifier for Climb and Swim checks. (Source: d20 PFSRD)
And from the official Paizo FAQ:
Tiny creatures, Climb, and Swim: Should all Tiny creatures use Dex instead of Str for Climb and Swim skill checks, or is that just for familiars?
Any creature of Tiny or smaller size should use its Dex
modifier instead of its Str modifier for Climb and Swim checks. This
will be added to Bestiary 4 as a new Universal Monster Rule. (Source: Paizo FAQ)
Though it is not explicit, this rule should apply universally to all creatures, not just monsters.