Are there any sources that discuss the significance/meaning of the fact that Elul and the Yomim Noraim almost immediately follow the 3 weeks of mourning? (actual sources preferred over speculation)
The Avodas Yisroel, the Koznitzer Maggid, in the end of Parshas Devarim has a piece about a new foundation being laid after Tisha B'av culminating in Simchas Torah.
Also, there is the Psikta which says that originally there was supposed to be a Yom Tov during every month of the summer, but Tammuz, Av and Elul lost out because of our sins. However they were paid back to us in Tishrei.
In the overview for Touched by Their Tears by Rabbi Yechiel Spero he says that the Zera Kodesh of Ropshitz writes that the 21 days leading up to Tisha Ba'av correspond to the days between Rosh Hashana and the end of Succos. Sefer Ma'or Vashemesh says adds that the 22nd day is Tisha Ba'av which corresponds to Shmini Atzeres.
There are several ways of understanding this. The way I see it now is that these days of feeling the loss of of the Beis Hamikdash and the distance from Hashem are supposed to make us want to come closer to Hashem, which we start with Elul, days of Ani Ledodi Vedodi Li, then the Aseres Yemei Teshuva, days of Kire'uhu Beheyoso Karos, then Succos when we enter Tzeila Demehemnusa, and finally Shmini Atzeres, Kasha Alai Preidaschem.