When I use a planetary time vortex does it reset the counter for advancement or just increase the development level? For instance, suppose I colonize/conquer a planet at 1500 hrs on a given day, establishing a New Colony. The next day I cash in AP and get a planetary time vortex at 1450 hrs. If I use the artifact at 1455, will I have just wasted the normal advancement to an "Emerging Colony" due to occur at 1500 hrs? Or will the development level be a "Developing Colony" after the artifact and standard advancement?

I havnt tested it on the level of scientific theory, but im very sure it just advances it.

I think I have answered my own question. Last night I was waiting for a planet that I colonized on Tuesday to click over from a new to an emerging colony. Eventually, I gave up as I couldn't recall when I originally took the planet. I used the planetary time vortex artifact and went to bed. This morning it is a developing colony. Therefore, it seems that the clock is not reset when using the artifact.

No it does not reset the clock, it just advances the colony's development status by one. Also if you want to veiw how long it's been since you developed a colony, cursor over the pop status area where it says calm, hostile, etc. It will tell you how long it's been since the planet was attacked. Aka: planet attacked 12 hours ago. Planet attacked 3 days 4 hours ago. Planet never attacked. This way you can see how long you have till the next development roll over

True, for conquered planets. Not true for colonized planets. If a planet has never been attacked, it says planet has never been attacked.