What you are asking about is, in terms of neuroscience and philosophy of mind, described as "the binding problem". For example, as stated in the article "Neuroscience: Toward Unbinding the Binding Problem" by David Whitney:
How the brain ‘binds’ information to create a coherent perceptual experience is an enduring question. Recent research in the psychophysics of perceptual binding and developments in fMRI analysis techniques are bringing us closer to an understanding of how the brain solves the binding problem.
The visual system is organized in a parallel, hierarchical and modular fashion. The distributed processing of visual information is thought to lead to an intriguing ‘binding’ problem: if the attributes of an object, such as a red car driving down the road, are processed in distinct pathways, regions or modules, then how does the visual system bind these features — color, shape and motion — consistently and accurately into a single unified percept (Figure 1A)?
Along with Subjectivity and Intentionality, John Searle, in section 3 ("Some Features of Consciousness") of his article "The Problem of Consciousness" describes Unity:
It is important to recognize that in non-pathological forms of consciousness we never just have, for example, a pain in the elbow, a feeling of warmth, or an experience of seeing something red, but we have them all occurring simultaneously as part of one unified conscious experience. Kant called this feature 'the transcendental unity of apperception'. Recently, in neurobiology it has been called 'the binding problem'. There are at least two aspects to this unity that require special mention. First, at any given instant all of our experiences are unified into a single conscious field. Second, the organization of our consciousness extends over more than simple instants. So, for example, if I begin speaking a sentence, I have to maintain in some sense at least an iconic memory of the beginning of the sentence so that I know what I am saying by the time I get to the end of the sentence. (boldface my own)
The notion of an "all or nothing" unity has its roots in DesCartes:
When I consider the mind, that is to say, myself inasmuch as I am only a thinking thing, I cannot distinguish in myself any parts, but apprehend myself to be clearly one and entire. 1641, p. 196
... and Leibniz (think "monad").
Another example of modern philosophical thinking in regards to the binding problem comes from Tim Bayne and David J. Chalmers, "What is the Unity of Consciousness?"
When I look at a red square, the color and the shape may be represented in different parts of my visual system. But somehow these separate pieces of information are brought together so that I experience a single red square, so that I can identify and report a red square, and so on. This phenomenon is often referred to as binding, and the question of how it is achieved is often referred to as the binding problem. The binding problem is in large part the problem of how objectual unity is possible.
Other than the article you referenced by Nagel, as for philosophical thought specifically regarding (what Searle would call a pathological form of consciousness) "split-brain":
Bayne and Chalmers consider brain bisection cases to be putative counter-examples because, on some concepts of the subject of experience, we can think of there still being one subject in these cases even though not all the conscious states are unified. There are at least three ways to respond. The simplest is just to deny that there is one subject, at least for the period of the split. A second would be to note that, however one counts subjects during the period of the split, there is evidence that many conscious experiences in that body are not like anything to some subject. If so, the apparent lack of conjoint consciousness of them will not be a problem. A third (advocated by Bayne and Chalmers, pp. 38–9) would be to urge that while there is clearly a breach in the unity of access consciousness (access to information for purposes of belief formation, behavioural control, and so on) during the period of the split, phenomenal unity may still extend across all the conscious experience.
It is noteworthy in the cases of Nagel, Whitney. Bayne and Chalmers that they do not distinguish "information" in the observer-relative sense which Claude Shannon uses from the sense which Searle identifies as first-person ontologically subjective and observer independent.