Any Alexa device can be made to play a custom spoken notification Or any of a number of sound effects, including a Siren, chimes, dogs barking etc. This is built into the official features of that device and is available through the “Alexa routines” feature.
You can integrate this with smartthings by using a virtual sensor. This works well when it works, but since the introduction of the second generation of the skill about a year ago it sometimes glitches where it doesn’t work for a few days or even a couple of weeks, and then it starts working again. Obviously that’s not very good for any kind of security system, since you never know for sure whether it’s going to work or not, but if you’re comfortable with that possibility, here’s the community FAQ on how to do that:
FAQ: Can I trigger an Echo Action without Speaking to It?
Google home does not yet offer any comparable method.
So since you already have an echo device, that would be the least expensive method for the spoken alert.
As far as how to integrate that with a delay for the ST siren, I will have to leave that to others to discuss. I’m pretty sure there’s a way to do that with the official smartthings features and may be another virtual switch, but I don’t know the details and I can’t myself use the app at this time because it is not voice navigable.
So hopefully others will chime in who can discuss the entire project with you. 