Ok, I did come up with this partial answer in terms of the response from noted figures in the UK after the first partition. Your question is quite complicated as during the three partitions, some MAJOR historical events happened (French Revolution, heads go rolling, so on and so forth....)

King George III's commentary: “The very extraordinary phenomenon of a coalition of the Courts of Vienna, Petersburgh, and Berlin to take what may suit their Separate conveniencies of the Kingdom of Poland, is so subversive of every idea of their mutual jealousies, and of the balance of Europe that it of necessary must give rise to very extraordinary Alliances amongst the other Powers"

King George III of Great Britain and of Ireland, Correspondence of King George The Third Vol. II, ed. John Fortescue, (London: Macmillan and Co, 1927), No. 1180

From the Annual Resister, a commentary on politics, edited and mostly written by Edmund Burke. This volume has been attributed to him:

“The present violent dismemberment and partition of Poland, without the pretence of war, or even the colour of right, is to be considered as the first very great breach in the modern political system of Europe. It is not (say the politicians of the continent) sapping by degrees the constitution of our great western republic, it is laying the axe at once to the root, in such a manner as threatens the total overthrow of the whole... We now behold the destruction of a great kingdom, with the consequent disarrangement of power, dominion, and commerce, with as total an indifference and unconcern...”

F.P. Lock, Edmund Burke: 1784-1797, (Oxford, Oxford University Press, 2006), 334

Now, they weren't particularly concerned with the well being of the Poles--they were much more alarmed at the shifts in power. The Kingdom of Poland was one of the larger empires in Europe, and frankly, stood as a nice buffer zone between Prussia and Russia...hence Davies calling it "God's Playground". The alliance between the Austrian, Prussians, and the Russians def. were a cause of concern. King George stated in the above sourced and quoted correspondence that the UK, the Netherlands, and the French should form their own alliance in order to shift the balance of power back. However, the issue wasn't discussed in Parliment.

Of course, the Polish response was the May 3rd Constitution to end the Noble Republic and to attempt to save Poland. This document was heavily influenced by Enlightened philosophy, and the US revolution. You can find it translated here

On May 9, 1791, The Times published an article titled “Revolution in Poland on Rational, Practicable, and Liberal Principles”. This article begins to speak of a wave of reform and change across Europe to adopt more liberal principles and policies and that in many cases, these newly adopted principles have largely benefited the people, but there are some cases in which the reforms have worked against the people. These radical interpertations of liberal principles are considered by the author of the article to be worse that the “most absolute monarchical despotism”. The negative consequences of some of these reforms were clearly in reference to to the situation in France. Although France was still technically a constitutional monarchy at this time, the radicalization was starting with the seizure of church lands and the banishment of convents, monanstaries, and the denial of authority to the Pope in the Civil Constitution of the Clergy

“Revolution in Poland on Rational, Practicable, and Liberal Principles,” The Times, May 9, 1791.

I don't read French or Turkish, so I'll have to do some digging to add the French and Ottoman responses.

I'm a Pole. The only official reaction by a foreign power I know of is the reaction of Ottoman Empire. In the Polish collective memory, Turkey is a country that never recognized the partitioning of Poland. I've heard that when Sultan met ambassadors, ambassador of Poland was always asked to show (some kind of official summoned ambassador of Poland during official ceremonies, even though there was no Polish amabasador due to partitions).

This is only a very partial answer, but Victor Hugo's Les Misérables (which takes place in 1832) describes a group of rather idealist Parisian students (the "friends of the ABC"), one of whom (Feuilly, the only one in the group who isn't a student — he's a self-taught fan-maker) is scandalized by the partition of Poland. The part describing this is volume III ("Marius"), book 4 ("the friends of the ABC"), chapter 1. The French original is here (search for "éventailliste"); here's an English translation:

Feuilly was a workingman, a fan-maker, orphaned both of father and mother, who earned with difficulty three francs a day, and had but one thought, to deliver the world. He had one other preoccupation, to educate himself; he called this also, delivering himself. He had taught himself to read and write; everything that he knew, he had learned by himself. Feuilly had a generous heart. The range of his embrace was immense. This orphan had adopted the peoples. As his mother had failed him, he meditated on his country. He brooded with the profound divination of the man of the people, over what we now call the idea of the nationality, had learned history with the express object of raging with full knowledge of the case. In this club of young Utopians, occupied chiefly with France, he represented the outside world. He had for his specialty Greece, Poland, Hungary, Romania, Italy. He uttered these names incessantly, appropriately and inappropriately, with the tenacity of right. The violations of Turkey on Greece and Thessaly, of Russia on Warsaw, of Austria on Venice, enraged him. Above all things, the great violence of 1772 aroused him. There is no more sovereign eloquence than the true in indignation; he was eloquent with that eloquence. He was inexhaustible on that infamous date of 1772, on the subject of that noble and valiant race suppressed by treason, and that three-sided crime, on that monstrous ambush, the prototype and pattern of all those horrible suppressions of states, which, since that time, have struck many a noble nation, and have annulled their certificate of birth, so to speak. All contemporary social crimes have their origin in the partition of Poland. The partition of Poland is a theorem of which all present political outrages are the corollaries. There has not been a despot, nor a traitor for nearly a century back, who has not signed, approved, counter-signed, and copied, ne variatur, the partition of Poland. When the record of modern treasons was examined, that was the first thing which made its appearance. The congress of Vienna consulted that crime before consummating its own. 1772 sounded the onset; 1815 was the death of the game. Such was Feuilly's habitual text. This poor workingman had constituted himself the tutor of Justice, and she recompensed him by rendering him great. The fact is, that there is eternity in right. Warsaw can no more be Tartar than Venice can be Teuton. Kings lose their pains and their honor in the attempt to make them so. Sooner or later, the submerged part floats to the surface and reappears. Greece becomes Greece again, Italy is once more Italy. The protest of right against the deed persists forever. The theft of a nation cannot be allowed by prescription. These lofty deeds of rascality have no future. A nation cannot have its mark extracted like a pocket handkerchief.

Would you mind sharing the citation for the cartoon you are referring to?

Was it not Polish-Lithuanian Commonwealth at the time of the partitions?