Most weather apps, including the stock weather, use data that comes from the National Weather Service relayed from airports. Weather information can be delayed at any point along the way. Some weather apps, like Weather Underground, use personal weather stations that are more localized and accurate. But they too can be slow. Opening the app itself can refresh the data. It also might help to toggle on Background App Refresh for the weather app. It will use a bit more battery, but could help to refresh your info.
Sent from my ancient but trustworthy iPhone 5.