There are tons of such apps [1] for a phone (which I assume you are asking about, not much use streaming from a laptop). However, the part I'm not having success with is finding an app that hooks into the voice recognition, and starts within a second of getting the command to launch, because both Android and iOS currently do not provide a way to certify and enable apps that can be launched from the lock screen without first unlocking.
The phone OS would provide specific services to keep such an app ready to launch at any moment, so it doesn't even need to be loaded from flash storage, and give it the ability to let the user set overrides on normal preferences like WiFi/cell signal to enable location services as soon as it starts, etc.
Elsewhere on the technology fronts, we need lower energy consumption for selectable lower-quality video and audio, to enable constant audio recording, and video when the phone detects the camera is out of a pocket/purse. Then the "...but you don't know what happened in the seconds leading up to the start of the video..." defense falls apart as non-HD-but-acceptable-quality-for-admissible-evidence becomes a matter of course. Some automatic rotation to save space, hook up to social media like Twitter to detect geographical hotspots of controversy to, or other measures to try to stymie gaming the rotation.
In the US specifically, but also elsewhere where it applies, we also need legislative overrides on two-party consent recording rules: any official, in any capacity, unless explicitly named in a national security letter or similar, receiving or benefiting from any public funding, is subject to one-party consent recording rules.
[1] https://www.google.com/search?q=live+stream+video+to+cloud&i...