Firstly, I would avoid dealing directly with the other party's insurer. Having the liable party be the paymaster of repair is always in conflict to your interest. Should things go wrong, you would have no rights to complain to the Financial Ombudsman Service; you are completely unprotected. You will need to inform your insurer when obtaining a new quote that you have been involved in a "non fault" incident. Though, in this situation, I would suggest either using a claims management company or claim via your insurer.