How are "vernacular" and "literary" readings of characters chosen and used outside of Mandarin?
Firstly, let us clarify one distinction: 


"literary Chinese" (文言文), a register of the written language in common use across the East Asian Sinosphere before the early 20th century. Its opposite is "vernacular Chinese" (白話文)
"literary readings of Chinese characters" (文讀), a set of readings mapping Chinese characters from written form to spoken form. Its opposite is "colloquial reading" (白讀).


The literary readings are derived from the correspondences to higher prestige variants, usually a form of Mandarin, from the historical capital at that time. They generally diffused into the local speech through education, and it is perhaps due to the meritocratic nature of the 科舉 (imperial examination) that this influence was exerted on "rural speech" at all.

Colloquial readings on the other hand arise from "native" sound changes, and are generally closer to Middle Chinese phonology (or even Old Chinese phonology in the case of the Min group) than the literary readings.

The exact nature of the literary / colloquial split varies across the varieties of Chinese, but it must be emphasised that each reading is generally fixed, and despite the "literary/colloquial" moniker does not correspond directly to linguistic register. Instead, the situation is much more similar to that of Japanese 音読み (on'yomi) vs 訓読み (kun'yomi), where on'yomi was borrowed from a form of Chinese, whilst kun'yomi was "native". However, there is little if any choice to use one or the other; the expression itself dictates which is used.

These factors include:


Names: for Min, given names are in literary, surnames are in colloquial. Country names are overwhelmingly in literary, especially if they are transliterated from other non-Chinese languages. Other geographical names are interesting, as there is a distinction (at least in Min Dong) between locations which are colloquial (nearer to the speech community geographically, hence in-group) and those which are literary (further). In Fuzhou Min Dong, 江 is colloquial gĕ̤ng in 閩江 Mìng-gĕ̤ng, the Min River, which flows through the city of Fuzhou; whereas it is literary gŏng in 江蘇 Gŏng-sŭ, referring to Jiangsu. 
Neologisms: anything perceived in the language environment as "modern" and therefore a loan from the prevailing standard speech will be in literary.
Bound vs free morphemes: most free morphemes are in colloquial, whereas bound morphemes in compound words have a higher probability of being in literary (although not always).  Occasionally, the same character's literary and colloquial readings will have spawned two lexemes with different meanings, e.g. Cantonese 驚 has the colloquial reading geng1 meaning "to be afraid, to fear", contrasted with literary ging1 "to frighten". But ging1 is found primarily in compounds whereas geng1 can be used independently.
Idioms: chengyu (成語) are in literary, but suyu (俗語) are generally in colloquial.


So I would expect 新加坡 and 新加坡共和国 to have only a literary pronunciation in Teochew (as well as all other Chinese topolects), and for that to be the only pronunciation used, both in formal and in informal settings. 

EDIT: I have seen the following claim (Rethinking East Asian Languages, Vernaculars, and Literacies, 1000–1919, p.279) - 


  in other [regional languages and dialects], speakers may shift between the two, though not always consciously; the choice is dictated by habit, local convention, and circumstances.


Though there is no a priori problem with the claim, the data used was a particular stylistic form of storytelling from Yangzhou, rather than general speech.