If no supersedeas bond or notice of appeal, as required of agencies exempt from filing bonds, has been filed and approved, the clerk of the court or justice of the peace shall issue the execution upon such judgment upon application of the successful party or his attorney after the expiration of thirty days from the time a final judgment is signed. If a timely motion for new trial or in arrest of judgment is filed, the clerk shall issue the execution upon the judgment on application of the party or his attorney after the expiration of thirty days from the time the order overruling the motion is signed or from the time the motion is overruled by operation of law.
Tex. R. Civ. P. 627