Though I haven't worked with Nuxt.js but general idea is that package returns "message" event which you can hook into. For example, here is vanilla JS that can be used:

<script>
var es = new EventSource("{{route('__sse_stream__')}}");

es.addEventListener("message", function (e) {
    var data = JSON.parse(e.data);
    alert(data.message);
}, false);

</script>

`

I assume you can do something similar with Nuxt.JS too to capture "message" event and react to it.