Why was there such a short, 8 Minute Tornado Warning for Atlanta, with the Weather Channel located there ?

Well, there is some misinformation in this question.

CNN is not affiliated with the Weather Channel. They are both based in the Atlanta area but the only thing that is common between them is the fact that they both appear on most cable TV systems.

Second, the Weather Channel does not issue tornado warnings. The Weather Channel only broadcasts the tornado warnings as issued by the National Weather Service. In this case, the National Weather Service Forecast Office located in Peachtree City is the office that is responsible for issuing the warnings for Atlanta.

The fact of the matter is that it is very difficult to issue warnings ahead of time. This was a fairly weak tornado (EF2) which likely formed rather quickly and from a storm that may or may not have been a mesocyclone. As such, it would be very likely that this storm may not have had the classical signature of a tornadic supercell and only when a strong circulation had formed and was detected by the Peachtree City radar, that the warning was issued.

Warning lead time is one of the major obstacles in tornado warnings. I believe the national average for tornado lead time is only around 13 minutes. The 8 minute lead time is actually well within current state of the art. It was only 30 years ago when the average lead time was actually a negative lead time. That is, the tornado was doing destruction before the tornado warning was issued.

This is also why the paying attention to the watches are so important. A very high percentage of all tornadoes are preceded by several hours by tornadoes watches which mean that the conditions are favorable for the development of tornadoes.