On the Japan Foundation website, you can download a PDF that outlines the differences between なか and うち. While that particular issue is not relevant to the question, the document explains all sorts of grammatical constructions using なか, including the「〜なかで」construction.
There are two distinct usages according to the document, which the author refers to as type A and type B respectively.
The first usage described in the document is that of type A. Here, the construction「〜なかで」can describe something such as an action, a state, an event, etc. within a general or broad setting that can give rise to that thing. The author refers to this broad setting as the frame. I'll try to illustrate what this means by clarifying the examples given in the document:
Throughout his book, the Shōbōgenzō, Dōgen provides detailed explanations about the concept of enlightenment in zen buddhism.
- Frame: The collection of works written by the Japanese zen buddhist monk Dōgen
- Within that frame: detailed explanations about enlightenment in zen buddhism
However, such high growth of the economy has brought about various problems within the Japanese society.
- Frame: the Japanese society
- Within that frame: various societal problems
The Sonnō jōi movement was established around the south-western clan domains under the Shogunate's rule.
- Frame: when the Shogunate was active
- Within that frame: a movement favouring the emperor and opposing the Shogunate was established
As you can see, the frame can be something well-defined (inside a book) or it can be a bit more abstract and vague (within society, under the administration).
While type A uses「〜なかで」to relate a certain frame or setting to events or actions that could occur within that frame, type B uses「〜なかで」more specifically to represent one or more options out of a range or a set of options. You're most likely more familiar with type B than type A, but I'll clarify with some more examples.
Nevertheless, France has the second lowest marriage rate in Europe after Sweden.
Range of choices: European countries
At the time, of all the children only one boy was to become head of the family.
Range of choices: the children of a certain family
To answer your question
There's another piece in the prologue preceding your screenshot. Putting both together looks like this
Since there is no clear set or range of choices, this usage of「〜なかで」is of type A. The thing that's happening is the last sentence:
There was a young boy who lived a carefree life.
Whereas the frame is contained in the previous sentence:
While a glimmer of hope for a better future budded among the citizens of London
Where the whole glimmer of hope stuff is contained in「その予感」I believe.
I'm not 100% confident in my translations quite yet, but I hope this helps!