Frankly, I think you will really struggle to find any books in the mathematician's style you exemplify that cover anything close to what I would call actual atomic physics. The closest you're likely to get is books on standard quantum mechanics for mathematicians, along the lines of Faddeev and Takhtajan's texts:
Both of these handle quantum mechanics, going all the way up to some fairly sophisticated QFT, in about as much rigour as is currently known for the results (at about the same tone as V.I. Arnold's Mathematical methods of classical mechanics).
However, they stick to standard quantum mechanics, and they do not go into the nitty-gritty of actual calculations for multi-electron atoms, messy additions of angular momenta, and all the mathematically-boring but experimentally-relevant stuff that makes up actual atomic physics. I see little point in a book that did such a thing, and I suspect most authors in a position to write one would tend to agree.
That said, I really don't see what it is you find so "mathematiciany" about Wiliams' Nuclear and particle physics (besides little boxes on the side with informal and heuristic explanations of the keywords, which add very little to actual rigour), and I don't really see how it is different to solid but standard textbooks like e.g.
or the other recommendations of Book recommendation for Atomic & Molecular physics.