Unlike on the App Store, there doesn't seem to be a way to lookup a TestFlight bundle in a non-authenticated manner.
Notice how the App Store lookup has no authentication:
http://itunes.apple.com/lookup?bundleId=com.example
The closest equivalents for TestFlight are:
https://testflight.apple.com/join/TEST_FLIGHT_ID
However, you can only see the name of the app and its icon in the HTML.
itms-beta://testflight.apple.com/join/TEST_FLIGHT_ID
Eventually hits an endpoint that seems promising, but it hits an authentication endpoint first, and passes along data from the authentication endpoint to the aforementioned endpoint (even for public betas).
Therefore, it doesn't seem like there is a way to publicly query TestFlight to learn about new versions. It seems to require authentication.