This question considers the respiratory apparatus of an aquatic (or amphibious) species descended from engineered humans.
The history of the species provides a relevant constraint. While the science in the setting is soft and the handwavery of fictional genetic engineering might be called upon to produce nigh arbitrary results, I would much prefer, if possible, to be able to claim “these structures that serve [some function] are actually homologous to [some other structure found in humans]!”, or “similar solutions are found in [some other species].”
Given these conditions, what would be the reasonable changes to make to humans such that they would be able to breathe underwater?
Relevant information come across during the search for an answer:
I recall seeing objections to the practicality of gills (on the neck/beneath the jawline) citing the square-cube law, even though I am unable to locate the source text. I am also uncertain about the specifics of this claim (might it still be possible for, say, large external gills to provide enough surface area?) — I’d be grateful if anyone could verify this or point me to a source. Another issue with modified humans with authentic gills is that they would likely have to lack ears, tonsils, and the thymus, since those are developed from the same structure that give rise to the branchial skeleton in fish, and it is unclear what impacts this would have on the rest of their physiology.
Some of the arguments presented above are also applicable to the option of having both gills and lungs.
Gills placed instead on the torso (to allow space for larger gills) is an often seen variation; and structures such as “gillungs”, as proposed in this answer, seem to be regarded as feasible. However, these appear difficult to justify from a developmental point of view (to have lungs connected to openings on the side of the torso involves a change in topology, and there would likely be complications associated with punching through the chest cavity?)
Finally, please note that this problem is not about how whale-humans are likely to be more practical than merfolk with gills: air-breathing aquatic humans exist separately in the same setting.