Hey /u/Wombfresh, this is a great question! Amusingly, I also just finished watching the Cunk on Earth series so this question feels doubly timely. I would agree that poverty and homelessness in Ancient Egypt were certainly different than how they are experienced presently in North American cities, but that doesn't mean they didn't exist.

Several years ago /u/XenophonTheAthenian answered a similar question about Ancient Rome and I think some of that insight holds true here:

The deal is that the vast majority of urban inhabitants were unskilled day-workers who lived a hand-to-mouth existence scrounging up whatever work they could. They survived by working on monumental building projects, at the Tiber dockyards, in the construction of roads and other manual labor, in minor unskilled suburban agricultural work, and in hawking. Their wages (provided they found work that day) were poor and paid daily. Harrison and others have pointed out that since rents were often paid daily and work was not always available in any given region of the city much of the city's population was probably essentially migratory, moving around in the city as working conditions changed. [...] Beggars, which we might consider the most abject of the poor, were habitually itinerant or homeless.

In the specific context of Ancient Egypt, Delphine Driaux recently published an article called "Toward a Study of the Poor and Poverty in Ancient Egypt: Preliminary Thoughts" which offers a nice overview of the topic. Driaux notes that poverty and homelessness remain much less studied topics in Egyptology than the life of the upper classes, but through an examination of textual, iconographic, and funerary evidence they are able to draw some preliminary conclusions.

Through a textual analysis of The Satire of the Trades Driaux begins to construct an image of one Ancient Egyptian view of the "poor." In the text a man named Khety encourages his son to become a scribe, while presenting the harsh realities of manual labour:

Nevertheless, although the text shows some condescension towards ordinary people, at no point is a term for ‘poor’ mentioned. That does not mean, however, that such workers were not considered part of the poor, since this text is dealing with the personal perceptions of a member of the dominant class, for whom undertaking manual labour alone is clearly a sufficient sign of a lower social condition. Khety also provides a detailed description of the external appearance and the physical condition of such labourers (the potter’s clothes are stiff with mud, the arms of the wall-builder are covered with earth, the gardener has a festering, swollen neck, etc.) to his son in order to persuade him to become a scribe, giving insight into what Egyptians thought were signs of deprivation. By explaining the negative aspects of these trades, Khety is presumably presenting here an Egyptian opinion on what is ‘poor’. (3)

Driaux then turns to the portrayal of labourers in iconography of the period. Extant images of labourers often portray them as balding, hairy, hunchbacked, unkempt, etc. Eventually, Driaux draws the conclusion that these iconographic portrayals:

have a sociological importance that implies a value judgement, which is rather negative. Perceived as being ‘different’ from the elites, these ordinary people are associated with negative or undesirable characteristics that can be turned easily into caricatures or stereotypes. (4)

But were these people - or at least some of them - homeless? It's very hard to say, though Driaux argues in favour of a so-called "archaeology of poverty" wherein members of the lower classes would live in simply constructed dwellings, which were often attached to larger buildings and under the auspices of a member of society higher up the social ladder. Using the sculptor Thutmose's workshop as an example, Driaux states that there were small brick dwellings built onto the back of his workshop's property that would have likely housed the labourers who aided him. These dwellings were organized around a large open space with a well in the centre. These men were the responsibility of Thutmose who was also responsible for providing them with food from his granaries. All of this combined leads Driaux to draw the conclusion that: "Poverty appears then as a social construction. It is indeed precisely by considering these individuals as inferior and different that the elites can categorize them as being poor and, as soon as this categorization is established, they can start to treat them as such." (12)

Driaux's ultimate conclusion is that the vast majority of the populace in Ancient Egypt lived in a state of "integrated poverty" (this includes tradespeople, labourers, the 10 men who assisted Thustmose, etc). Driaux is careful to note, however, that some words seem to indicate a different kind of poverty:

Nevertheless, it appears that some words (like ḳrj) cover the idea of a ‘marginal poverty’ that takes in a very small portion of the population who probably lived on the fringes of the society. Another group or degree of poverty that one can describe as ‘integrated poverty’, more visible in the funerary iconography, consists of the vast majority of the ancient Egyptian society—the peasants, manual workers, etc. The many aspects of poverty of these individuals can be due to the precariousness of their social status, their lack of material resources, but also the privation of power or influence, professional skills, knowledge or even physical capacities. Amongst the mass of poor, one must therefore take a nuanced understanding, based not only on social status, but also on criteria which differentiate the groups of poor within society. (14)

So, was there homelessness in Ancient Egypt? Almost certainly. As with Ancient Rome, however, it seems that the majority of those in something akin to poverty would have been living in "integrated poverty" but when housing and food are tied so strictly to the whims of a member of the elite there was almost certainly iterant homelessness between jobs, while looking for work, etc. even if that was not the norm.

SOURCES

1. Driaux, Delphine. “Toward a Study of the Poor and Poverty in Ancient Egypt: Preliminary Thoughts.” Cambridge Archaeological Journal, vol. 30, no. 1, Feb. 2020, pp. 1–19, doi:10.1017/S0959774319000258.