I move unit A to battlefield A and conquered it. Then I use Cataclysm to move my Jarvan IV, Royal Heir from base to battlefield B. Battlefield B has Riven, Broken Blade defending it. The opponent uses Siren's Song to move my unit at battlefield A to battlefield B as a reaction to my Cataclysm. Will a new combat showdown starts before or after Cataclysm resolves?

No new showdown starts before Cataclysm resolves — the Combat Showdown opens after it, once the Chain is empty.

Siren's Song is legal as a Reaction (Cataclysm's "opponents can't play spells this turn" only applies once Cataclysm has resolved), and it resolves first, moving your unit from battlefield A to battlefield B where Riven is defending. That applies Contested and gets a Combat staged, but the Chain still holds Cataclysm, so the turn isn't in a Neutral Open state (Rule 310.2) and the showdown just sits staged (Rule 323.11).

Then Cataclysm resolves, moving Jarvan IV into battlefield B. In the cleanup that follows resolution the Chain is empty, so the turn player initiates the staged showdown — and because units controlled by both players are present, it opens directly as a Combat Showdown/Combat (Rule 466.1).

One consequence worth noting: you are the Attacker, because the first unit to apply Contested to battlefield B was your unit A (moved by your opponent's own Siren's Song), so Jarvan IV enters the combat with the Attacker designation and Assault's +1 [M].