Looking at the original publication of
S. Nussbaumerianum, you can read (Google Translate is helpful for those of us not fluent in German) Bitter's description published in February 1923.
https://www.biodiversitylibrar...
On p. 283, you can see that Bitter did wonder about the association with Hamet's
S. adolphi, but he notes that Hamet's description is incomplete—giving, for example, no info about the color of the leaves and flowers—and in some cases his description does not fit Bitter's plant. I'm not sure how much I can rely on Google Translate, but it seems that Hamet felt that his
S. adolphi was closely associated with
S. allantoides, and Bitter did not think his plant had a close relationship with
allantoides.
Perhaps Hamet described his plant first, but he apparently didn't describe it very well if Bitter (while acknowledging that Hamet must have had inadequate specimens) enumerates points where Hamet's description of
S. Adolphi did not fit that of Bitter's
S. Nussbaumerianum.
I don't know who later decided that the plants were synonymous (or why), but Bitter had Hamet's description in hand, and was familiar with the Sedums brought back from Mexico by Purpur in 1907 (& from other trips?) and grown in several German gardens. Perhaps they were not obviously synonymous, but some nomenclaturist felt that Hamet's description needed to be shoehorned into synonymity with one of Purpur's Mexican Sedums.
For me, it raises the question of whether naming priority should be given to an earlier description which is incomplete, and apparently wrong or ambiguous about key points. I am assuming that Bitter wasn't faking his doubts about Hamet's plant, and that Hamet's description was indeed not very accurate/complete.