In the Witcher, what is the future of the Continent according to the books?

The "Encyclopaedia Maxima Mundi" (a Nilfgaardian encyclopedia written centuries after the events of the series) references these events has having taken place in "the Empire’s northern regions". Thus giving a good implication that the northern kingdoms were taken.

Delannoy, Flourens, linguist and historian b. 1432 in Vicovaro, in the years 1460– 1475 secretary and librarian to the imperial court. Indefatiguable scholar if legends and folktales, he wrote many treasises considered classics of ancient language and literature of the Empire's northern regions. His most important works are: Myths and Legends of the Peoples of the North; Fairy Tales and Stories; The Surprise, or the Myth of the Elder Blood; A Saga about a Witcher, and The Witcher and the Witcher Girl, or the Endless Search. From 1476 professor at the academy in Castell Graupian, where d. 1510.

Effenberg and Talbot, Encyclopaedia Maxima Mundi, Volume IV pg. 267, The Tower of the Swallow (UK edition)

The relevant passage from "Encyclopaedia Maxima Mundi" is about Ithlinne Aegli's prophecies:

(...) przepowiedziała I. Wojny Północne (1239–1268), Wielkie Zarazy (1268, 1272 i 1294), krwawą Wojnę Dwóch Jednorożców (1309–1318) i najazd Haaków (1350).

Which can be translated as "Itlina fortold Northern Wars (1239–1268), Great Plagues (1268, 1272 & 1294), bloody War of the Two Unicorns (1309–1318) & Haak invasion (1350).

That means in books' continuity there was no Third Northern War, present in games, (or hypothetical fourth one) and peace was stable, but people had bigger problems with plagues - not a good time for warfare.

Another passage from Encyclopaedia describes Kerack as being a part of kingdom Cidaris and no mention of any servitude to Nilfgaard, just that kingdom of Kerack was partitioned by its neighbors.

To answer the question first: No Nilfgaard did not win the continent.

Jarre (from the temple of Melitele) was eyes witness in the last big fight of the Nilfgaardians army. As he is an old man he writes down this:

So lag also die Macht Nilfgaards auf den Feldern von Brenna in Schutt und Asche, und dem Vormarsch des Kaiserreichs nach Norden wurde schließlich Einhalt geboten. An Gefallenen und Gefangenen verlor das Kaiserreich bei Brenna vierundvierzigtausend Mann. [...]So also wurde Brenna zum Anfang vom Ende. [...] Es muss daran erinnert werden, dass Jan Natalis [...] nach Süden vorstieß. Eine Kavallerieeinheit unter Adam Pangratt und Julia Abatemarco zerschlug zwei Divisionen der Dritten Armee, die Menno Coehoorn verspätet zu Hilfe kommen wollten, zerschmetterte sie derart, dass nec nuntius gladis. Als die Nachricht davon den Rest der Heeresgruppe »Mitte« erreichte, suchte diese schändlich das Weite und wich eilends über die Jaruga zurück, und da Foltest und Natalis ihnen auf den Fersen folgten, verloren die Kaiserlichen den ganzen Tross und sämtliche Belagerungsmaschinen, mit denen sie in ihrem Hochmut gedacht hatten, Wyzima, Gors Velen und Nowigrad zu erobern. Und wie eine Lawine, die von den Bergen herunterkommt, immer mehr Schnee mit sich reißt und größer wird, hatte auch Brenna immer gravierendere Folgen für Nilfgaard. [...] Als aber de Wette von Brenna erfuhr, als ihn die Botschaft erreichte, [...] ließ er augenblicklich zum Rückzug blasen und floh Hals über Kopf über den Fluss nach Cintra, [...]. Nur in Nastrog, Rosrog und Bodrog, den nicht eroberten Festungen, blieb eine starke Besatzung, die erst nach dem Frieden von Cintra dort in Ehren und mit den Standarten abzog. In Aedirn indes führte die Nachricht von Brenna dazu, dass die verfeindeten Könige Demawend und Henselt einander die Hand reichten und gemeinsam gegen Nilfgaard ins Feld zogen. [...] Verstärkt durch Einheiten aus Redanien und durch Königin Meves Guerillakämpfer, [...]. Demawend und Henselt aber griffen unverzüglich die Nilfgaarder an, und dort, bei Aldersberg, wohl gemäß der historischen Gerechtigkeit, zerschlugen sie sie in einer wackeren Schlacht, obwohl Nilfgaard noch immer erheblich in der Überzahl war. Also pflegen Geist und Kunstfertigkeit über dumpfe und brutale Gewalt zu triumphieren. (achtes Kapitel "Die Dame vom See")

He remembers the fight near Brenna and that this was the beginning of the end of the nilfgaardian army. They have to go back behind the Jaruga-line. The last part of the army was destroid near Aldersberg by Demawend and Henselt. The following agreement got the name "peace of cintra". So in the 7th book (The Witcher on Wikipedia) the last I know is that Nilfgaard does not own the whole continent until Jarre will die as old man (what the mention of his grandchildren suggest). But I have not read "Season of Storms" yet, and because Ciri is not bound to the "actual" time, it is possible that after Jarre died there could be change. Like in every world...

translated via translation software:

So the power of Nilfgaard was in ruins in the fields of Brenna, and the advance of the Empire north was finally halted. The empire lost forty-four thousand men killed and captured at Brenna. [...] So that's how Brenna became the beginning of the end. [...] It must be remembered that Jan Natalis [...] advanced south. A cavalry unit under Adam Pangratt and Julia Abatemarco smashed two divisions of the Third Army that wanted to come to the aid of Menno Coehoorn late, so shattered them that nec nuntius gladis. When the news of this reached the rest of Army Group Center, they shamefully ran away and hurried back over the Jaruga, and with Foltest and Natalis following on their heels, the Imperial troops lost the whole entourage and all of the siege engines they were using had thought in their arrogance to conquer Vizima, Gors Velen and Nowigrad. And just like an avalanche that comes down from the mountains, pulls more and more snow with it and grows larger, Brenna also had increasingly serious consequences for Nilfgaard. [...] But when de Wette found out about Brenna when the embassy reached him, [...] he immediately let the blow to retreat and fled headlong across the river to Cintra, [...]. Only in Nastrog, Rosrog and Bodrog, the fortresses that had not been conquered, remained a strong garrison, which only withdrew there in honor and with the standards after the Peace of Cintra. In Aedirn, however, the news from Brenna led to the warring kings Demawend and Henselt shaking hands and going together against Nilfgaard. [...] Reinforced by units from Redania and by Queen Meves guerrilla fighters, [...]. Demawend and Henselt, however, attacked the Nilfgaardians immediately, and there, at Aldersberg, probably in accordance with historical justice, they crushed them in a brave battle, although Nilfgaard was still considerably in the majority. So spirit and craftsmanship triumph over dull and brutal violence. (eighth chapter "The Lady from the Lake")

(Maybe someone could help me with the english parts of the book?)