[Vuejs]-My fetched data doesn't render in my template in nuxt 3

Your variable allDataWeather is not reactive, you need to make it reactive, using ref or reactive Vue methods.

const allDataWeather = reactive([]);

// your logics
allDataWeather.value.push(yourNewWeatherValue);

Also, note that <script setup> is async by default, thus you can use await inside without hacks.