Others with more experience may be along later with a better answer, but since you seem to be in distress I'll try for a start:

Where did you miss your flight (what city are you flying out of) and where were you when you encountered the car trouble (ie, engine wouldn't start in your driveway, flat tire on the highway)? If I was flying out of an airport with CX flights departing after mine, like HKG, TPE, other places with multiple services, I probably would try to get to the airport if I could as soon as possible, if it's not hugely far from where you are. Assuming there are flights departing after yours, CX staff would still be around and you could try to talk to them directly and see if they could rebook you on a later flight. Even if it's inconvenient, it's better than having to wait on the phone for a long time given all the problems people are having getting through to an agent.

If you're flying from an airport with only one flight per day and you've missed that one, or if you live hours away from the airport, your best bet is probably to call as soon as their office opens in the morning.

A question for the experts -- does CX operate a flat-tire rule (sometimes airlines are lenient with change fees if they're convinced you tried to get to the airport on time but something went totally wrong) and if so, would the OP have to head to the airport asap to plead for a break on the no-show and/or change fees?

Anyway, OP, this is one person's opinion so you might also want to wait for some other experts to chime in with advice before deciding what to do.