If a creature is grappled, its speed becomes 0. If it escapes that grapple, does it regain its normal speed as soon as it escapes the grapple or on its next turn?
Each combat round consists of a standard action, a move, and any bonus actions that it can do.
To escape Grapple, that's a standard action. That means the PC still has a move it can do.
So yes, it can move as soon as it escapes.
My take. 
Your speed is only affected by the Grappled condition while you are under the Grappled condition - your speed returns to normal immediately upon you no longer being Grappled.