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As for your question, presuming the player with Uril cast the STP you can Deflecting Swat it to Uril. That player is the owner & controller of the spell and Deflecting Swat doesn't change that.
Comprehensive Rules 112.2.
A spell’s owner is the same as the owner of the card that represents it, unless it’s a copy. In that case, the owner of the spell is the player under whose control it was put on the stack. A spell’s controller is, by default, the player who put it on the stack. Every spell has a controller.
Opponent targets my creature with Swords.
I Deflect the Swords targeting Uril.
Is this correct and legal?
As for your question, presuming the player with Uril cast the STP you can Deflecting Swat it to Uril. That player is the owner & controller of the spell and Deflecting Swat doesn't change that.