I don't believe there is a direct integration with Twilio, but you might want to check out this post where some others are discussing other options for sending SMS notifications from Bamboo.
ugh! That is a problem. The only other suggestion I can make at this point would be to look at this Bamboo feature request. I know that it is request SMS notifications for a manual step, but it might be worth adding your specific use case which (I think) would make this capability more useful.
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