This doesn't work with Twilio Functions check @philnash answer

You have to return the callback first to answer the call, and then in a delayed promise send the SMS

exports.handler = function (context, event, callback) {
    const delay = (delayMS) => new Promise((res) => {
        setTimeout(res, delayMS)
    })

    // Create a new voice response object
    const client = context.getTwilioClient();
    const client_number = event.From;
    const twilio_number = +xxxx;

    let twiml = new Twilio.twiml.VoiceResponse();
    twiml.say("");   // respond to voice caller
    callback(null, twiml);

    return (
        delay(10000)
            .then(() =>
                client.messages.create({  // Send SMS
                    to: client_number,
                    from: twilio_number,
                    body: "Hello on SMS"
                })
            )
            .then(() => console.log("SMS sent"))
    )
}

Hopefully this helps.

Note: delay is an utility function where you pass the amounts of milliseconds you want to wait.

Twilio developer evangelist here.

This is not something you can do with just a Twilio Function. While aLittleSalty's answer will work in a situation where your application and event loop continues to run, like in an Express app that you run, but in a Twilio Function once you call the callback method the event loop is terminated so the delay will be cancelled.

Similarly, Twilio Function execution time is limited to 10 seconds, so you can't delay a message by a couple of minutes even if the above would work.

One idea could be that you send the message once the call has completed. Your example code only includes a <Say> so I will work with that, but you could do this in other ways if the call continues differently.

First, update your initial Function to return just the TwiML, but add a <Redirect> after the <Say> so that once the message completes Twilio sends a new webhook.

exports.handler = function (context, event, callback) {
  let twiml = new Twilio.twiml.VoiceResponse();
  twiml.say("");   // respond to voice caller
  twiml.redirect("/after_say");
  callback(null, twiml);
};

Then create another Function at the /after_say path that sends the message and hangs up the call.

exports.handler = function (context, event, callback) {
  // Create a new voice response object
  const client = context.getTwilioClient();
  const client_number = event.From;
  const twilio_number = +xxxx;

  let twiml = new Twilio.twiml.VoiceResponse();
  twiml.hangup();

  client.messages.create({  // Send SMS
    to: client_number,
    from: twilio_number,
    body: "Hello on SMS"
  }).then(() => {
    callback(null, twiml);
  }).catch(error => {
    console.error(error);
    callback(null, twiml); 
  });
}

That's still not going to give you a couple of minutes between the call coming in and the message being sent, but it will hopefully send once the call is complete and be a better experience.