Is the Hadith, "Go back and pray, for you have not prayed," to be acted upon after entering the mosque and before greeting the congregants, or does it include not greeting anyone from home until the mosque?

There is no harm for someone going to prayer in greeting those he meets on the way; rather, it is commendable. As for prioritizing the greeting of the mosque over greeting those present, this occurs when one enters the mosque, where the greeting of the mosque is performed first because it is a right of Allah Almighty, and then one greets the people. Scholars have differed on this issue: Ibn al-Qayyim held the view of prioritizing the greeting of the mosque absolutely, while Ibn Rajab was of the opinion that whoever enters the mosque and passes by people therein should greet them first and then pray. All of this is commendable, not obligatory.