Just use Javascript. There’s no need to do this on the server.

<div id="time-div" data-time={Timex.from_now(time)}>&nbsp;</div>
<script>
  const div = document.querySelector("#time-div");
  const startTime = new Date(div.dataset.time);

  function updateTime() {
    const now = new Date();
    const diff = Math.floor((now - startTime) / 1000);
    div.textContent = new Date(now).toLocaleString();
  }

  setInterval(updateTime, 1000);
  updateTime();
</script>

My preferred approach to display relative times or localize timestamps accordingly to the user browser preferences is to use the relative-time-element web-component.

You just need to:

<relative-time datetime={time} />

<div class="text-xs font-normal">
  {time
  |> DateTime.shift_zone!("Europe/London")
  |> Calendar.strftime("%d %B %H:%M",
    month_names: fn month ->
      {"Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", "Dec"}
      |> elem(month - 1)
    end
  )}&emsp;-&emsp;
  <relative-time
    datetime={time}
    format="elapsed"
    threshold="PT60M"
    formatStyle="narrow"
  /> AGO
</div>

20 Sep 19:35 - 22m 27s AGO