I threw together a quick prototype using a NodeMCU, small oled display and a gps unit. Then spent some time working with the software.
I had seen a number random crashes and found that this was due to power demands from the GPS module so I added a large-ish capacitor nearby.
Since the ESP8266 only has one USART I’m using a software serial implementation. I've still seen some crashes and decoding the stack trace they are happening on the interrupt service routine for this module. A quick look at the modules code shows a WAIT macro that may not be safe in an interrupt service routine,
I'm also using a nice, small NMEA parsing library to parse the GPS messages.
Its responding to ntp requests and syncing with the PPS signal from the GPS unit. I am seeing a few times where the NMEA message from the GPS is parsed after the next second pulse from the GPS. Because I'm validating validating that the time I have is correct with each message I see it time warp back one second then timeworn forward one second a few times a day. I think that the ESP wifi housekeeping functions may sometimes delay the serial parsing. Or that rendering the display could be taking too long to render each second.
While I don't have a GPS module, I do have boards from SynchroClock that have the DS3231 real time clock 1hz routed to a GPIO pin on on the ESP8266. This lets me pretend I have a GPS (and I read the current time from the DS3231).
I used a strategy mentioned by @Nick Sayer, in the comments on one of his projects here, and used the ESP8266 builtin CPU cycle counter, 80Mhz, to interpolate the time between seconds. And it answers queries!
flugelberry$ ntpdate -q -u 192.168.0.29 server 192.168.0.29, stratum 1, offset 0.273729, delay 0.06538 3 Nov 19:39:51 ntpdate: adjust time server 192.168.0.29 offset 0.273729 sec
EDIT: It turns out that ESP.getCycleCount() is not very consistent. Using this I end up with up with well over 250us of jitter! Maybe something with idle sleep or some other power saving mode? In any case using micros() works much much better with jitter less than 20us!