(Obv I know it's more complicated than that, but where else can you be reductionist if not the internet)
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With a bare minimum implementation you should be able to get the GPS time from the NMEA strings. You just need to guarantee you'll have GPS lock at least intermittently, which is probably true for a fleet of monitored cars in nearly all situations.
As for bringing that into a computer, I'm having a hard time imagining a hardware setup where a software PLL won't give you millisecond-level accuracy (assuming you have an internal clock with at least that much precision). 1 second accuracy implies just using the timestamp from the last NEMA string with no further processing...
1. That's only true with a super, super naive approach. You can easily estimate the average latency as part of your prototyping, and subtract that (while "live") from when you receive the full NEMA string. That will knock down the (in)accuracy by at least an order of magnitude, probably two, depending on latency variance. NEMA messages are (almost[1]) always sent synchronously with the PPS.
2. Again I'm having trouble imagining this naive hardware setup where you don't have a better timestamp than 1 second. NEMA usually comes over a UART, so just timestamping the first byte of the NEMA string will give you ~millisecond accuracy. Even a cheap USB 1.1 UART dongle that only polls at 125Hz should give you ~10 millisecond accuracy. As above, you can estimate average latency during development to improve accuracy during usage.
You'll need to do a little bit of software filtering to remove outliers, of course, but that requires no extra hardware.
The point is: The technical bar to getting 10ms accuracy with no special hardware and no PPS is basically zero.
[1] I'm hedging with almost, but I've never seen a counterexample.
This is a fundamental system for the company, so it's worth cracking a few heads over the issue.