Well, in regards to your question, there should be a distinction made between the general idea of eastern origins and its importance, and that of the exact Uralic origins of the Hungarian language. Also in advance, since English isn’t my first language, sorry for language mistakes.

The fact that Hungarians came from the east, and at the time of their conquest could be included in the broad and historically continuous group of (semi-)nomadic Steppe peoples, appears in some of the earliest known self-representations of Hungarians. In early Hungarian chronicles (Gesta Hungarorum by Master P., ca. 1200; Gesta Hungarorum et Hunnorum by Simon of Kéza, ca. 1283; Chronicum Pictum by (most likely) Márk of Kált, ca. 1360) there is an attempt at establishing continuity regarding various eastern people groups known in Antique and Medieval Western canon, notably the Scythians and Huns, and the conquering Magyars. An important part of it was the descendance of the ruling Árpád dynasty from Attila himself. Thus the Hunnic origin became a very integral part of the self-image of the Hungarian nobility. It was and is up to much debate that the Hunnic origin was A: simply appropriated from the general view Western chronicles, in which the various nomadic conquerors of Central and Eastern Europe were lumped together due to (sometimes superficial) cultural similarities, and made into a glorious past by Hungarian chroniclers or B: if there was some degree of a similar tradition among the Magyars or the Árpád dynasty to begin with, which was the precursor of said idea, and became "Westernized" and "Christianized" (for instance there was great effort made in the chronicles to find biblically inspired ancestors too for Hungarians, most notably Nimrod, the great hunter) to become the Hunnic tradition of Hungarian origins. Conquering Magyars were also depicted in these sources with similar clothes and material culture as later nomadic peoples (notably Pechenegs and Cumans) who had a mostly antagonistic relationship with the Kingdom of Hungary, so the cultural similarity of Magyars to groups of the Eurasian steppe was understood.

Nevertheless which origin the Hunnic-Scythian tradition was, there is a stark difference between the representation of Huns in other European sources and Hungarian chronicles, as in the latter there is a sense of pride from descending from the "scourge of God" and overall Atilla and the Huns are presented in a positive manner (though there can be parallels made with Polish Sarmatism, in which Polish nobility claimed origins in nomad Sarmatians, but as far as I am aware it had a much smaller scope in time than the Hunnic tradition). As I said this became of very important in the self-image of Hungarian nobility, and it remained a defining part of it well into later centuries. For example in the 17th century Nádasdy Mausoleum, a collected biographies of "the chieftains and kings of Hungarians" sponsored by the Nádasdy aristocratic family, and published first in Nürnberg, 1664, several mythological and semi-mythological Hunnic chieftains are present and Attila is very prominently featured, proudly proclaimed “orbis flagellum” i.e. “scourge of the world”.

Why am I telling you this? You asked about the specific Uralic origins and its effect on Hungarian national consciousness, not the general understanding of the Eastern origins of Hungarians, right? Well, but the thing is, in order to understand how the uncovering of the Uralic origins of the Hungarian language affected the Hungarian national sense of self, it is instrumental to appreciate into what circumstances did it arrive. The fact that Hungarian is different from other European languages was established relatively early. In general, humanists of the 16th century liked to link Hungarian to a language of high esteem, Hebrew. There were sporadic reports of Hungarian resembling this or that Uralic languages. For example Leibniz grouped Hungarian with Finnic languages in his work of Brevis designatio meditationum de originibus gentium ductis potissimum ex indicio linguarum. Philip Johan von Strahlenberg, a Swedish military man also noticed similarities between Finnish and Hungarian (Das Nord-und Ostliche Theil von Europa und Asia, 1730), and in the 18th century the idea of Turkic origin also gained traction, especially considering its fitting into the axiom of the Hunnic past (notably Hungarian scholar György Pray in his Annales veteres Hunnorum, Avarorum et Hungarorum). But these were all isolated attempts to establish some kind of relationship based on perceived parallels and not systematic scientific inquiries.

Maximilian Hell, a Jesuit priest and astronomer went on an expedition to the northern shore of Norway in 1769. With him was another Jesuit János Sajnovics. It was him who noticed that Sami and Hungarian have an affinity in terms of basic vocabulary. Even though he had many ideas that would be considered far from anything resembling modern linguistic analysis (most notably questionable etymologies and also far-fetched attempts at related languages, like Chinese), he brought the idea of Hungarian being related to another language that was later considered part of the Uralic language family into the Hungarian public consciousness.

You may have noticed how I referred to the Hunnic tradition as an axiom. Because an axiom it was. It was basically an important basis of Hungarian self-image, and as the meaning of the “Hungarian nation” started to shift from meaning exclusively the nobility of Hungary to a broader, more “modern” sense of nationhood, the significance of this glorious past stayed important. To give a relatively basic rundown of the reactions of the Hungarian public, I tried to group said reactions, I grouped them in 3, but please note that in reality these groups overlapped somewhat and the clear distinction between them is for the sake of argument and illustration.

Sámuel Gyarmathi was a Hungarian nobleman and medical doctor from Kolozsvár (present-day Cluj-Napoca, Romania), who at first was appalled by the idea Sajnovics presented. He initially started to examine several Uralic languages to disprove their relatedness to Hungarian, but quickly realized that they are in fact seem to be in the same language group. His Affinitas (published in Göttingen, 1799) is generally considered the first systematic analysis of the possibility of an Uralic language family and thus the establishment of modern Hungarian linguistics. But his admittance was the exception rather than rule.

A second kind of acceptance existed, that had given in to the idea of Finno-Ugric relatedness of Hungarian, but only with it being made fit into the Hunnic tradition and the more accepted Turkic relations. For example Pál Beregszászi Nagy, another early Hungarian linguist, in a presentation given in the University of Erlangen in 1795, said that an acceptance of Hungarian being related to “Finnic languages” would only be possible if there would be a link established between them and Turkic languages. An interesting “pop-culture” rendition of the novel idea of the Finno-Ugric language group was András Dugonics’s Etelka series. In which the 9th century Hungarian noblewoman visits the Hunnic relatives in the northern country of “Karjel” (Karelia).

But the most common reaction to this new idea was either passive ignorance or active disdain. The most common argument was probably the dislike for these ‘nothing peoples with no history’ or as Mór Jókai, one of the most famous 19th century Hungarian writers put in his short story Mahizeth (1852): “And then they had children, and those were such dwarfs, such miserables, as if not the same God made them as the others.. A four feet man among them is a giant, their ladies are ugly, the younglings are strengthless, their faces numb...Love and jealousy are unknown among them, they offer their ladies to strangers..And all nations on Earth can find in this miserable people’s language reminiscent memories, but most the Hungarian… They have no history, no ancestors. They don’t even remember their own father’s names. But as if the world would like to scorn them too, they are still called “smouldering ones” (“lappanó”). They live at the edge of the world, they are the very last ones among humans.” Also as I mentioned there was the ignoring it. For example there was an examination of all Hungarian history books for pupils between 1770 and 1850. Almost all of them mentions Hunnic origins. Some of them features the kind of fantastical, grossly exaggerated kind of discourse, that was connected to István Horvát and linked Hungarians to several ancient civilisations through naive etymologies. Only one of them even mentioned the possible relation to what will be established as Uralic languages. Ézsaiás Budai in Magyar ország históriája (Debrecen, 1805) says that: “Some contemporary scholars...seem to deny the kinship of Hungarians with the Hunnus and the Avar only to bring them to Finnus roots”.

In the first half of the 19th century the direction of exploration and inquiry was still often defined by looking for Inner Asian roots. For example the author of the first Tibetan - English dictionary, Sándor Körösi Csoma, went on being one of the first Western explorers of the Himalayas, but he did so, because he was looking for the relatives of the Hungarians. (Continuing in the comments below)