Edit: the question was actually about a website being able to trigger some kind of sound, alert or notification while the phone screen is off, not about keeping track of the time. That is not possible due to permissions (thankfully).
Apps have access to additional permissions that might make this feasible to do as an app.
Original answer basically answering "How to measure elapsed time in a browser on mobile from clicking a button even when screen is off or browser is closed":
What I would do is as soon as the timer is started you store the timer start time in localstorage and then use a setInterval to update the UI based on your stored value. I would make this accurate to the second to avoid a very heavy UI update cycle. Something like this works (tested on Android / Opera)
(Cannot use a snippet due to localStorage allow-same-origin)
<div>-</div>
<button>Start</button>
<script>
const out = document.querySelector("div")
const btn = document.querySelector("button")
let timer, startTime;
btn.addEventListener("click", () => {
// If timer is started, reset
if (timer) {
clearInterval(timer)
}
startTime = Date.now()
localStorage.setItem("startTime", startTime)
timer = setInterval(() => {
const now = Date.now()
const millis = now - Number(localStorage.startTime)
const seconds = Math.floor((millis / 1000) % 60)
const minutes = Math.floor((millis / 1000 / 60) % 60)
out.innerText = minutes + ":" + seconds
}, 1000)
console.log("Started interval timer", timer)
})
</script>
You could make it even work between browser shutdowns by checking the localstorage during page load
