yh will be in your name and you will have to pay the excess if the third party denies the accident ever happened, its good u got the reg number and make and model, as this is all what insurance needs, police will be useless as usual.
get the witness details for sure, and when the third party does not admit fault or denies it totally, then there will get statements from the witnesses make sure the witness includes make model and reg plate, thats where the witness for my dads went wrong.
you got a good chase, but if you are in a position to pay the excess get the car repaired, if the third party admits then you sorted, car will be repaired and you will be offered a decent hire car, unlike the ford ka i got. as there recover the full costs, if there deny then prepar yourself to put up with the damage on the car if you not able to pay the excess, and the excess will only be waived once it settled.
good luck