i would check your policy paperwork , it sounds to me like the insurance maybe not following your agreement.
having also got fully comp insurance and having been in this situation myself . i found that contacting the insurance and advicing off the accident , i didnt have much paperwork to do , just a simple statement of what happened. " if the other party denies fault this may lead to other paperwork ,
if your entitled to a courtacy car during through the insurance, your normal insurance would usualy be transfered to that car for the duration, and only the normal charges would apply in the case of an accident. " check this regarding any paperwork you have".
there should never unless there are legal issues to sort be any need to bring in ny third party company to make a claim.
as for costs incured " ie : voluntery excess !" this is usualy claimed back from the other persons insurance automaticly " if they are at fault " you will normaly have to pay the voluntery excess however should recieve this back from your normal insurance company when they recieve payment from the other insurance company.