Hi,
Hard to anything definitively without a bit more info. What is the 'Weewx website', is that the Standard skin or Seasons skin or something else? What weather station are you using? Are you posting to WU using rapid fire?
Typically the Standard and Seasons skins will be updated with each new archive record as is comes in. So if your archive interval is five minutes you will see an update every five minutes and the wind gust data on the page should reflect the highest wind speed seen in the previous archive interval (five minutes). WU can be updated using rapid fire or via normal archive records, the difference being rapid fire uses loop packet data from your station which tends use more of a 'point in time' value whereas non-rapid fire uses data from the archive record which should match what you see on the Standard and Seasons skins. How frequent rapid fires updates are sent depends on your station and how often it emits loop packets. Of course, we have no idea what WU does with your data once received. WOW on the other hand is updated with archive data but instead of being updated every archive interval the default update period is every 15 minutes. So if you had a five minute archive interval only every third archive record would be sent to WOW and it would be quite conceivable to see different values on WOW for up to two-thirds of the time.
One thing you could do is set debug = 2 in weewx.conf and then restart WeeWX. That will cause WeeWX to log the data being sent to WU and WOW. You could then see how the posted data compares to what is displayed on your site. Bear in mind that the units used in the data posted to WU and WOW is US customary so you may need to do some conversions if your site displays Metric.
Gary