Doesn't appear to be great sources out there for some of this info! (But then, that's what StackExchange has great promise in being!)
In the US, this manual for the official instrument setup, as well as this old manual on how to take observations gives quite a bit of good detail on how wind observations are done...
In the US, wind speed is the most recent 2 minute average.
A thorough explanation for gusts for ASOS, which is about as readable as syrup:
In the manual procedure, a gust is reported when an observer sees
rapid fluctuations in sensor wind speed indications with a variation
of 10 knots or more between peaks and lulls during the 10-minutes
before the observation. The reported gust is taken from the maximum
“instantaneous” wind speed observed during this period. The
average 2-minute wind is used to report wind direction and
wind speed. Conceivably, an average 2-minute wind speed as low
as 3 knots (observed in the last minute) may be reported
with a gust of 10 knots (observed in the last 10 minutes).
Observations of 5 knots with gusts of 10 to 15 knots, however, are the
more common minimum values reported.
The ASOS algorithm also relies on a 10-minute observation period to
determine gusts, but uses it in a different way. Once every 5 seconds,
the ASOS computes the greatest 5-second average wind speed (and
corresponding direction) during the past minute, and once each
minute stores this information in memory for 12 hours.
Once every 5 seconds the ASOS computes the current 2-minute average
wind speed and compares it with the greatest 5-second average
wind speed during the past minute. If the current 2-minute average
wind speed is equal to or greater than 9 knots and the greatest
5-second average wind speed (during the past minute) exceeds the
current 2-minute average speed by 5-knots or more, then the greatest
5-second average speed observed during the past minute is stored in
memory as a gust for 10 minutes.
Once every 5 seconds, the ASOS compares the highest gust stored in memory
for the past 10 minutes with the current 2-minute average wind speed.
If the difference between the two is 3 knots or more, the current
reported wind speed is greater than 2 knots, and the highest gust
exceeds the minimum 5-second wind speed in the past 10 minutes by 10
knots or more, then the highest gust stored in memory is designated as
the reportable gust. This value is appended to the current wind
direction and speed reported in the OMO, computer-generated
voice messages, and the METAR/SPECI reports. The minimum gust speed
reported by ASOS is 14 knots. Wind speeds from 3 knots and 11 knots
may be reported with gusts to 14 knots. For example, a 2-minute
average wind of 240 degrees at 10 knots with gusts to 20 knots is
reported as:
“24010G20KT.
Of course, the wind speed rules aren't standardized globally, as this WRAL meteorologist discusses. In fact, they aren't even entirely consistent in the US, as the Hurricane Center uses a 1-minute average for sustained winds for hurricane intensity, and 3-5 second period for gusts.
So, even having done coursework on meteorological measurement systems, it's about as muddled to me now as it is to you. Probably report the most recent 2 minute average as the actual wind, and the highest instantaneous or 5-second average as the gust? (with requirement that it exceeds set criteria as DavePhD's answer alludes to, if you wish)
