The effect of the Russians taking Galicia depends heavily on whether it occurs in 1859 or 1866.
It is, perhaps LEAST likely that Russia taking Galicia in either 1859 or 1866 would bring about the same alignment of Austro-Hungary+Prussian dominated Germany against France+Russia+(half heartedly) Britain with an eventual blowout in the early 20th century. Not that I'm a fan of butterflies but this would be taking thing too far.
let's consider the different scenarios:
1859-
Napoleon proposed more than Russia taking Galica- he favored supporting Hungarian independence. This is only a decade after the crushing of the 1848 rebellions and much of the Magyar elite is unreconciled to collaboration with Austria so Austrian collapse on both fronts might well lead to this outcome. Of course, there were several reasons why Russia chose not to intervene in this war. The top three were:
a. Fear of Prussian intervention following the coming to power of william I after his more conservative and pro Russian brother's heart attack.
b. The final phase of the Chechen and Circassian wars (the first would end in 1859 and the latter in 1864), dragged out to to the insistence of Dmitri Milyutin of making a finish of it by ethnically cleansing the Circassians. This conflict tied up a third or more of the Russian military budget.
c. Ongoing administrative and millitary reforms.
d. concern that collapsing Austria might disturb the balance of power with Prussia and lead to German unification.
In my TL I had the heart attack delayed, and the reformers convinced that "a short victorious war" would help establish their credbility and carry out reforms, leading them to offer a more generous deal to the Circassian and Chechen insurgents, mobilize, and enter the war after Napoleon III wins a more desicive Soliffino. Hungary secedes following a Russian incursion across the Carpathians and the death of Franz Joseph when he tries to break out of the siege of lake Garda leads to a peace which limits the Austrian empire to Bohemia, Moravia, the Austiran crown lands, and Croatia-Slavonia. Bismark heavy handedly backstabs and annexes the minor states of Northern Germany, but Bavria et al remain independent and form a southern confederation.
Italy is united in a loose French sponsered confederation under the titular presidency of the Pope, and Hungary is strongarmed into electing a Russian monarch under a regency council- yes, that's Alexander III. I left future German unification an open ended question.
However, to be quite honest, all of the above was gilding the lilly a bit too much in favor of the Russo-French alliance and somewhat railroading affairs into a similiar outcome as OTL- just earlier
More realistically, a POD similiar to what I proposed would probably involve the more Russophile William Fredrick receive some kind of Russian/French reassurance of support for paramouncy in the post war German confederation, and perhaps some minor territorial acquisitions from Austria (Austrian Silesia). I suspect that Russian intervention in Galicia would lead to Austria losing Hungary and Venetia, as well as Lombardy. Hungary would likely elect a Scandinavian, or British, prince as a monarch.
Prussian post war paramouncy in the German confederation might pass unchallenged, leading to a Prussian presidency, and a confederation which includes Prussia's portion of the Polish partition, and alots it more votes in the federal assembly (which was heavily weighted in favor of the smaller states until dissolution) and a Prussian dominated federal army which gradually comes to eclipse the state army. Still, the resulting federal structure is very, very, weak. Both Germany and Austria spent a smaller proportion of their national income on their millitary prior to WWI as compared to Russia and France and underutilized their manpower potential for political reasons stemming from the federal structures of each. I would suspect that a German confederation such as this would be even less prepared for war TTL, if something analogous broke out in the early 20th century and all else is equal. However, all else is not equal.
Prusso-French War in 1860?
First of all, it is possible that Prussian paramouncy IS challenged and that France (and possibly Russia) view their pre-war reassurances to Fredrick William as non-binding. If push comes to a shove and France ends up supporting the minor states against Prussia (WHich Napoleon III might, given greater success in Solofino and Prussia's presumed reluctance to pay him off with the left bank of the Rhine) then the millitary outcome of an earlier war, prior to the kicking in of the Prussian military reforms, in which the southern states (and Hanover and Saxony and Rump Austria) are fighting against Prussia, rather than with it would probably end in very limited Prussian gains in which they might gain Hannover but might be forced to cede the left Bank of the Rhine and forced to recognize a southern German confederation.
PanGerman-Russo-French war in 1864?
Alternatively, Bismark might be smart enough to head this off by using the Shelzig-Holstein dispute to pull off something similiar to what he did OTL- unifying German public opinion behind him and corraling the rulers with it. France alone will probably not intervene against in favor in Denmark under this scenario. Whether Russia does depends heavily on whether the Polish rebllion of 1863 takes place as OTL- if it does, Russia probably doesn't. If it does, then Bismark probably backs dows with some kind of Brith mediated face saving formula.
Franco-RumpAustro-Prussian war in 1866?
The most likely outcome of the war is that Bismark puts Schlzeig-Holstein under the rule of the federal assembly, rather than seeking to annex it to Prussia in order to provoke a war as he did OTL. If he does, than the whole issue is avoided and the confederation remains a loose, albeit economically and socially consolidating mishmash. If push comes to shove as OTL in 1866 in the post war manuvers between Bismark and Austria/minor states, then things look very different. I imagine France WOULD intervene against Prussia. But 1866, even if it is not 1870 is not 1860 either. France is unlikely to do very well against the reformed Prussian millitary and the minor German states and Rump-Austria. You would probably have a remaining southern block, but Prussia would probably keep the left Bank of the Rhine and annex Saxony (whose ruler might be compensated with Austrian Bohemia), as well as Hannover and Kessel.
A silly thing in the Balkans?
Either of these outcomes likely avoids Russian-Austrian conflict in the Balkans and the 1878 Austrian annexation of Bosnia. If the Circassian ethnic cleansing is averted this also likely delays the spontaneous Slavic rebellion against the Ottomans. It is most likely that war, when it comes, is a matter of Russia moving in to sweep the Ottomans away, and Prussia/North Germany, possibly in alliance with Piedmont, moving in to complete their national unification while those powers are otherwise engages, leaving France to oppose this. in all scenarios save the final (war in 1866)- and probably fail, since in absence of defeat in 1870, it is unlikely to reform it's army on the Prussian model. Russia is also likely to lose a drawn out water against the UK if it occurs prior to 1920 or so- the UK simply outproduces it too badly. There is no reason, however, to assume that the Franco-SouthGerman-Prussian and Russian-British-Ottoman conflict need be joined into the same war- it seems more likely that the victor in whichever conflict ends first (Prussia, possibly France if war was fought in 1866 an dlessons drawn), mediates a resolution of the other conflict.
1866-
If Russia actively enters into the Austro-Prussian war this occurs post the 1863 Polish rebellion. indeed, denying future rebels a home base in Galicia might be the reason for their entry. Hungary, in this scenario, is unlikely to break away unless Bismark fails to restrain Fredrick William. If he doesn't than a Franco-Prussian war in 1866 seems likely with a similiar outcome to that described above (Statues quo on the Rhine, Prussian annexation of Saxony and Hannover, consolidation of loose South German confederacy).
If he does restrain him, which seems more likely, then the pragmatic compromise likely unfolds as OTL, but Hungary has more power in the relationship and is probably able to gain greater budgets for the Honved. Assuming the Franco-Prussian war proceeds as OTL (which seems likely), that Austria occupies Bosnia in agreement with Russia in 1877, the question then arises as to whether Bismark would choose to support Britain and Austria in pushing minimizing Ottoman losses in 1878. I guess he probably would and that the gradual consolidation of rival Austro-German and Russo-French blocks occurs as OTL.
A related question is whether Russian involvement in 1866 poses enough of a menace to Austria so that Italy's performance in the war is less underwhelming, enabling them to claim Italia Irrendetia. If it does then Italy probably remains in the Austro-German camp.
FInally, the question is what touches off WWI. My guess is that Hungarian secession is more likely to touch it off than conflict with Serbia. 1906 was a very near thing.
Still, if all other things are equal and a war IS touched off around 1914 with Italy still clamoring for Italia Irendete, then the opening phases of WWI look very different. Whoever is holding the passes, Russia can probably play defense against Austria while focusing on Germany. Germany is likely to realize this, and opt to play defense against France while pushing RUssia to the Vistula or the Bug- which the Russian war plans OTL called for not contesting until mobilization was complete and the French assaulted the German Western front. Absent a Gorlice Tarnow Russia will be able to put up more of a fight in 1915 and 1916- but fewer Gemrans bogged down in the West means more Germans fighting in the East. Also, a common (German) command of the Easter front will be in place from the get go with Austria playing a second fiddle (and likely trouncing the Serbs earlier). ANd Britain is less likely to intervene, at least in 1914 and by land.
Expect a Russian collapse (but no Bolsheviks) earlier rather than later and a German protectorate/personal union in Poland and the Baltics accompanied by a staues quo agreement, in the West.
If Germany, in a brain dead move still pivots through Belgium then... well, it's screwed. It will face four, rather than two armies in the opening battles, and probably abandon Eastern prussia with it's grain surplus and manpower. The Russians aren;t marching on Berlin, and probably fail to make much of an incursion into Silesia. indeed, they are probably pushed back to the Vistula in 1915. But there is no Great Retreat and the Austro-Geermans face a siege which they probably lose in 1916.