Assuming better weather, it'll recover. And be careful now about when to cut - the ground will take longer to dry out than the grass, which depending on where you are and your ground may make (a) travelling difficult for the contractor's machinery and (b) it more difficult to get the crop dried as it will be lying on wet ground.
Around here we sometimes have to leave the grass in its cut rows, not spread out, while the wind dries the ground between the rows, then spread and row up again in the in-between spaces to let the wind dry where the rows were last time.