My understanding is that if the chauffeur fails to get you to the airport in time for departure the airline will class you as a no-show.
If you have allowed more that what could be considered sufficient time for the journey from your location to the airport you may have grounds to recover your costs from the chauffeur company, but I expect that there is something in the T's & C's that will cover this.
This situation happened to a friend of mine who was in a car travelling to Gatwick for a flight when the car was involved in an accident on the motorway. As a result of the delay he missed his flight. Fortunately his travel insurance picked up the tab for costs he incurred since the chauffeur company refused to do so.