> Yes I am referring to NMEA string usage because that's as good as you'll get if the PPS output is not connected.
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.