First off, Prussia still retained a sizable portion of its gains from the Partitions. Contrast this map of the Partitions and this of the post-Napoleonic Poland and it is clear Prussia still retained Posen and West Prussia, which the Hohenzollerns had acquired during the First and Second Partitions. Although these regions did have a fairly large German-speaking minority, a number of Polish-speakers remained under Prussian rule and the existence of Polish minority would be a vexing problem for Prussia and the united Germany through to the twentieth century.
Nonetheless, a good portion of the postwar Kingdom of Poland (often called Congress Poland) did come from the Prussian portion of the Partitions. This was partly a result of the Napoleonic client state, the Duchy of Warsaw itself conformed to much of the Kingdom of Poland's own boundaries. Emperor Alexander I occupied the Duchy in the course of 1813 and the Russian Army took over de facto control of the Duchy's institutions. Although there were a great many Polonophobes within the tsarist state apparatus, especially within the military, Alexander I did envision himself as a champion of a resuscitated Polish kingdom under direct Romanov control. One of the influential ministers in Alexander I's entourage, Prince Adam Czartoryski, was an emigre Pole and pushed Alexander I to back a Polish kingdom as a liberal experiment that would be pro-Russian. This political move for an enlarged Russian sphere in Poland dovetailed with the immediate needs of the war in Central Europe, so much of Russian power was focused on what had been Prussian Poland as opposed to Galicia, which was far from the front.
Prussia was not in much of a position to argue with the Russians laying claims to their territory from the Third Partitions. The need to defeat Napoleon trumped a number of other concerns in Prussia's immediate foreign policy and the evolving coalition politics meant that King Frederick Wilhelm III was very much dependent upon Romanov support. Not only did he face a potential regional rival in the form of the Swedish King Jean Bernadotte, who joined the coalition against Napoleon, but Prussia was the most vulnerable of the Coalition powers. Prussia alone of the the Coalition was a battlefield during the 1813 campaign and had Napoleon triumphed after the 1812 fiasco, Prussia's betrayal of France at Tauroggen in December 1812 could have had severe consequences. Thus Prussia stood the most to lose of the Coalition parties so it was not in Frederick Wilhelm III's interest to rock the boat over Poland.
Moreover, Alexander I also dangled the prospects of Prussian enlargement into Central Europe as a way to sweeten the deal over Poland. Prussian statesmen pushed for a dismemberment of Saxony and compensations in central and Western Germany as its just reward for the defeat of Napoleon. This itself was more attractive to Berlin than Poland; after all, Saxony had a largely Protestant German-speaking population, unlike the mostly Catholic, Polish-speaking areas around Warsaw. The Romanov's support for Prussian claims in Germany was part of its negotiation tactics to gain leverage for the Polish Kingdom. While Metternich was loath to see a stronger Kingdom replace the Duchy, he was concerned that Prussian enlargement would also undercut Austria's own position within the emerging German Confederation.
The ultimate settlement of the Polish question at the Congress of Vienna was not so straightforward as a fourth partition. Piotr Wandycz has aptly noted that one of the paradoxes of the Congress was that it " both reaffirmed the Partitions and recognized the existence of the Polish nation." The Kingdom had its own currency and army, as well as being staffed by many veterans of the Napoleonic Duchy. But these measures never fully satisfied Polish national sentiments nor did they eliminate the Polonophobic attitudes of the Russian state, which only grew over time. Moreover, the fortress regions of Poland were very hotly contested among the Partitioners. The Prussians managed to retain Toruń/Thorn which helped it protect Posen and in exchange for the Prussian claims on Leipzig. Austria also pushed for the creation of semi-independent microstates duchy of Zator, duchy of Oświęcim and Żywiec, as well as the Free City of Krakow. Not only did these semi-independent states prevent their fortresses from falling into either Prussian or Russian hands, but the Duchies' membership in the German Confederation allowed them to act as a tripwire against Russian expansion of the Kingdom.
Alexander I's dalliance with Polish nationalism and a liberal kingdom proved to be short-lived. But this imperial interest in Poland under Russian suzerainty dovetailed with the immediate needs of the war effort to control Poland and bring the Russian war effort into Central Europe. Thus Russia was able to set the rough boundaries of a postwar Poland, which included a good chunk of Prussia's share of the Third Partition. Berlin for its part did not find this territorial rearrangement as a negative since it had promises of gaining German territories in exchange for ceding claims over central Poland. Not only was this more attractive to Prussia, the Hohenzollerns were also in no position to argue the point if they were so inclined.
Sources
Kraehe, Enno E. Metternich's German Policy, Volume II. Princeton: Princeton University Press, 1983.
Vick, Brian E. The Congress of Vienna Power and Politics After Napoleon. The Congress of Vienna. Cambridge: Harvard University Press, 2014.
Wandycz, Piotr S. The Lands of Partitioned Poland, 1795-1918. Seattle: University of Washington Press, 1996.
I have just been reading about this actually. The answer is that Russia actually acquired this territory during the Napoleonic Wars.
You have to remember that the Napoleonic Wars were not one single long conflict (though it often seems that way in English-language histories because for Great Britain, it pretty much was) but a series of distinct wars fought across Europe by Napoleon. For most European countries the wars were separated by extended periods of peace, until another coalition assembled to try to push back French power.
The Second and Third Partitions of Poland took place in the early phases of the French Revolutionary Wars and are quite closely tied to it. Prussia (which had previously pursued a policy of alliance with France to counterbalance Austria) agreed to join Austria in the War of the First Coalition(1792-1797), conditional on Austrian acceptance of Prussian territorial ambitions in Poland. This war was supposed to be an easy landgrab due to France's revolutionary turmoil, but a combination of unexpectedly strong resistance and coalition incompetence led to early defeats.
At the same time, the Russians had invaded Poland to put down a rebellion there. The Prussians saw an opportunity, and asked for a share of Polish territory as compensation for their efforts against France (which the Russians were keen on because they feared revolutionary contagion). They got it in the Second Partition of Poland (1793). After this, Prussian involvement in the Coalition waned, because they'd got what they wanted out of the war and were leaning back toward their old policy of seeing France as a counterbalance to Austria. In 1795 Prussia signed a separate peace with France, and took advantage of another failed uprising in Poland to carve the remainder of it up between themselves, the Russians and the Austrians. The elimination of Poland was arguably a big geopolitical mistake, because it had been a convenient buffer protecting Prussia from the east - from then on, relations with Russia would be massively important to Prussian and later German history.
Prussia was then at peace for 11 years while Napoleon took power in France and fought the Wars of the Second (1798-1802) and Third (1805) Coalitions. Prussian policy wavered back and forth during this time, reflecting both the difficult situation they found themselves in as by far the weakest of the Great Powers, as well as a tug-of-war at court between ministers favouring siding with France and siding with Russia. Prussia hoped to leverage peace with France to absorb northern Germany into their sphere of influence, but didn't reckon with Napoleon's habit of breaking almost every agreement he signed. By late 1805 the Prussians were running out of patience and leaning toward the Third Coalition, so they sent a minister to deliver an ultimatum to Napoleon. He arrived in Napoleon's camp several days after the Austrian and Russian armies had been crushed at the battle of Austerlitz, recognised that this was perhaps not the best time, and instead concluded a non-agression pact with France.
Napoleon's actions after Austerlitz demonstrated increasing contempt for Prussian interests, and Prussian anger grew and grew until the War of the Fourth Coalition began in 1806. Though Austria and Russia also joined in, they were still recovering from Austerlitz. Initially, Prussia had to face the might of the French army alone and were totally humiliated at the battle of Jena-Auerstedt. Berlin fell only 19 days after the invasion began, and the king was forced to flee and spent the rest of the war with the remnants of his army, fighting as a decidedly junior partner to the Russians.
The Russians put up a much more convincing fight, but were eventually forced to sue for peace, which was settled at Tilsit, on a (very large) raft in the middle of a river while the two armies encamped on opposing sides. To illustrate how terrible the Prussian king's situation had become, he was also left on the riverbank while Napoleon and the Tsar negotiated the future of his kingdom. The terms were very harsh for Prussia. On the one hand, they lost swathes of territory in the west to Napoleon's German allies/puppets, had to accept French garrisons in various places, and paid a huge indemnity. On the other they lost everything that had been gained from the Second and Third Partitions of Poland, and even some from the First Partition. Some of this land was given to Russia, and the rest became the Duchy of Warsaw - essentially an "independent Poland" set up to be a French ally. So that's what happened to Prussian Poland - bargained away to Russia by Napoleon at the Treaty of Tilsit.
The situation was straightened out a little at the Congress of Vienna after Napoleon had been deposed. The Duchy of Warsaw's annexation by Russia was basically a foregone conclusion, though it was nominally reorganised into "Congress Poland" under personal union with Russia. However, Prussia was compensated with a little bit of its lost territories in the form of the similar "Grand Duchy of Posen."
Source: Iron Kingdom: The Rise and Downfall of Prussia 1600-1947 by Christopher Clark
