tl;dr although basenjis are an "ancient" dog lineage and although they live in an area that has never been part of wolves' geographic range, they are thought (like all other dogs) to have originated in Asia by splitting off from wolves, after which their ancestors migrated to Africa sometime around 15,000 years ago.
Domestic dogs are thought to be monophyletic, i.e. all modern domestic dogs are more closely related to each other than any is to a non-dog canid species (although there is some amount of genetic admixture due to hybridization with other canids); even papers that propose multiple domestication events (Frantz et al 2016) still show domestic dogs as being monophyletic. Another way of saying this is that basenjis are descended from the "most recent common ancestor" (MRCA) of all domestic dogs, which in turn is most closely related to wolves.
Larson et al 2014 say that
three of the ancient breeds (Basenjis, Dingoes, and New Guinea Singing Dogs) come from regions outside the natural range of Canis lupus (the dog’s wild ancestor) and where dogs were introduced more than 10,000 y after domestication. These results demonstrate that the unifying characteristic among all genetically distinct so-called ancient breeds is a lack of recent admixture with other breeds likely facilitated by geographic and cultural isolation. Furthermore, these genetically distinct ancient breeds only appear so because of their relative isolation.
The combination of introgression and bottlenecks suggests that basal breeds have little or no genetic connections to their ancestral populations, and that genetic distinctiveness alone cannot be used as a proxy to signify an ancient heritage.
In other words, all dogs are thought to have split off evolutionarily from wolves in Siberia around 23,000 years ago (Perri et al. 2021). The ancestors of basenjis came to central Africa [on their own or accompanying humans] sometime after that; because their ancestors were more isolated from other dog breeds, they appear to have split off from the lineage containing "modern" breeds a long time ago.
(From Liu et al. 2018:
Although genome-wide analyses have estimated that a subset of ancestral dogs migrated toward the Middle East, Europe, and into Africa ~ 15,000 years ago (Wang, Zhai et al. 2016), the oldest archaeological evidence for African dogs was found in Egypt, dating ca. 6300–5600 BC (Mitchell 2015).
In other words, we don't actually know very much about how dogs got from Eurasia to Africa.)
Phylogeny from Larson et al 2012:
Although the paper corresponding to the news article you cite (Edwards et al. 2021) is an impressive genomic/bionformatic advance (high-quality genome of basenjis), it doesn't actually say much about the evolutionary history of basenjis.
Edwards, Richard J., Matt A. Field, James M. Ferguson, Olga Dudchenko, Jens Keilwagen, Benjamin D. Rosen, Gary S. Johnson, et al. “Chromosome-Length Genome Assembly and Structural Variations of the Primal Basenji Dog (Canis Lupus Familiaris) Genome.” BMC Genomics 22, no. 1 (March 16, 2021): 188. https://doi.org/10.1186/s12864-021-07493-6.
Frantz, Laurent A. F., Victoria E. Mullin, Maud Pionnier-Capitan, Ophélie Lebrasseur, Morgane Ollivier, Angela Perri, Anna Linderholm, et al. “Genomic and Archaeological Evidence Suggest a Dual Origin of Domestic Dogs.” Science 352, no. 6290 (June 3, 2016): 1228–31. https://doi.org/10.1126/science.aaf3161.
Larson, Greger, Elinor K. Karlsson, Angela Perri, Matthew T. Webster, Simon Y. W. Ho, Joris Peters, Peter W. Stahl, et al. “Rethinking Dog Domestication by Integrating Genetics, Archeology, and Biogeography.” Proceedings of the National Academy of Sciences of the United States of America 109, no. 23 (June 5, 2012): 8878–83. https://doi.org/10.1073/pnas.1203005109.
Liu, Yan-Hu, Lu Wang, Tao Xu, Xiaomin Guo, Yang Li, Ting-Ting Yin, He-Chuan Yang, et al. “Whole-Genome Sequencing of African Dogs Provides Insights into Adaptations against Tropical Parasites.” Molecular Biology and Evolution 35, no. 2 (February 1, 2018): 287–98. https://doi.org/10.1093/molbev/msx258.
Perri, Angela R., Tatiana R. Feuerborn, Laurent A. F. Frantz, Greger Larson, Ripan S. Malhi, David J. Meltzer, and Kelsey E. Witt. “Dog Domestication and the Dual Dispersal of People and Dogs into the Americas.” Proceedings of the National Academy of Sciences 118, no. 6 (February 9, 2021). https://doi.org/10.1073/pnas.2010083118.