Hi,
You need to press the 'claims handling company' you were referred to progress this asap for you. I'm also thinking why your actual insurance company referred you to the accident claim handling company. From what you've said, each party is blaming the other for the incident and so each party's insurance company should 'talk' to look at the responses sent in by you and the 3rd party to see if they can settle it either against the 3rd party, or you. Once liability either way has been decided, the claim should then move quicker.
Usual stuff then applies, if the incident is settled against the 3rd party, then your insurance company will authorise repairs to your vehicle etc and claim back the costs from the 3rd party's company. Your no claims bonus won't be affected as such etc, although annoyingly, the incident will be 'noted' on your record and it's 'likely' your insurance premium will rise when it comes to renewal time as they'll see the incident that's been listed. If the incident is settled against you, then it's the opposite way around and the 3rd party's company will claim off your insurance company for damage to their vehicle. Your insurance company will then fix your car via your policy and the incident will be noted on your record as a 'fault' accident on your record and your no claims discount will reduce and your insurance premium renewal is likely to be significantly/proportionally higher.
You'll need to keep talking to the 3rd cog accident claims company to push the claim and thus resolution forward and keep maintaining your view it was the 3rd party's fault.
Cheers, DtM! West London & Slough UK!