Easiest low-impact POD: Bismarck dies of the infection that nearly cost him his leg when he was ambassador to Russia in 1857-1861.
That way, in 1862, when the still rather new King of Prussia Wilhem I clashes with the increasingly liberal Landtag, the option to make Bismarck the Minister President of Prussia isn't there to be chosen. Wilhelm in OTL believed that Bismarck was the only man who could get things done on the King's terms. Before the plan to appoint Bismarck was suggested (by Roon), Wilhelm had been of a mind to abdicate in favour of his son. In a wotld without Bismarck, I imagine this ends up happening. (There is no other prespective champion that Wilhelm felt he could rely on, in OTL.)
Friedrich III becomes King of Prussia, over a quarter-century before he (very briefly) became Emperor of Germany in OTL. The liberal-tending Landtag has its liberal-tending Monarch. Of course, Friedrich opposed his father's abdication in OTL, calling it "
a threat to the dynasty, the country and the Crown". He wouldn't be very happy with the politicians that drove his father to it. Yet as far as politics were concerned, he also explicitly stated that he was fully in agreement with what he called "
the essential liberal policy for internal and foreign affairs".
Come the 1863 elections, and the liberals get a clear majority (as in OTL), and surely a liberal Minister President is appointed. The course of Prussian politics is forever altered. Indeed,
@ATP's "Prussians will be Prussians" remark is off-target here: it ignores how decisive the Bismarck period really was. Without Bismarck, there is no deliberate conflict between the conservatives and the progressives, and we instead get a King (and a Cabinet) dedicated to a "middle way". There is no
Kulturkampf, either. The typically Bismarckian policies of "Germanising" the Polish lands, for instance, would not take shape. Instead, you'd see the more traditional attitude of neglectful dismissal. (With the consequence, presumably, that far fewer Germans are settled in Polish lands.)
Friedrich III would most probably diminish the OTL power of Roon and Moltke (who in OTL worked together with Bismarck), because they'd be too conservative for his tastes. He may even remove them altogether. Either way, the militarisation of Prussian society
and the re-organisation of its armed forces would be deflected by this change, to a significant degree. Note also that Friedrich was an avid opponent of antisemitism, and would do whatever he could to steer society away from that impulse.
When Frederick VII of Denmark dies in late 1863, Prussian public opinion will still lean heavily towards Augustenburg's claim, as in OTL. This ATL's Prussia, however, will have good relations with Britain, and will understand that the British interest is to keep the Straits in the hands of minor power, and out of the hands of a major power. Friedrick III will most probably use his connection to Britain in order to foist a diplomatic deal on Denmark: namely that Christian gets Denmark and gets to integrate Schleswig, but in return has to give up Holstein
altogether. (Presumably, the Augustenburg line gets that, ruling it as a member state of the German Confederation, divorced wholly from Denmark.)
The fact that there is no Gastein Convention also handily removes the whole point of dispute that Bismarck used to spark the Austro-Prussian War in OTL. Without this war, there is also no secret alliance with Italy. Although the Italians wish to claim the "unredeemed" lands of Veneto and Trieste, they can't face the Austrians alone-- and these regions remain in Austrian hands. (Conversely, Nice and Savoy aren't ceded to France.)
Speaking of Austria: the German Confederation, formally under Austrian leadership, continues to exist. Indeed, Friedrich III was sympathetic to the
Großdeutsche Lösung. Far more than either his father or his son, he seriously entertained the idea of restoring the united Mediaeval empire, at least in some form (beyond the nebulous form of the Confederation). On the other hand, he'll eventually have to face that there is no chance for Prussia to become the hegemon within such a structure. If it is attempted, none other than Austria can be the carrier of the Imperial legacy. This will still cause tension. Not war, but nevertheless an obstacle to any "Greater German" ambition.
Yet Austria will try, as it did in OTL, and probably with better chances for some kind of success. Initially, the greater opportunity for Austria to assert itself (and Prussia's failure to do same) will surely strengthen the Austrian position. Over time, this will inevitably start to spook Prussia, and the other Protestant states of Northern Germany.
I actually think that the greater nationalist impulse will still drive things forward, and that matters won't stay "frozen" in a tableau of many "little Germanies". Instead, I think that increasing Prussian opposition (albeit
polite opposition) to overt Austrian hegemony will result in a Prussian camp and an Austrian camp. All others will have to choose. The ultimate dividing line will end up along the OTL border of the North German Confederation-- which I suspect will still coalesce, albeit later and more peacefully. The Southern states will fall into the Habsburg orbit, recognising the Imperial authority.
My estimation is that the ATL North German
Confederation will instead be the North German
Federation, a true "Bundesstaat", albeit one dominated by Prussia. (But a bit less so than in OTL, because Prussia lacks various war gains.) The Habsburg realm, meanwhile, may quite plausible address its own lingering issues by reforming into a (restored) Holy Roman Empire. One that, following the Mediaeval example, is not strictly bound to German lands. As such, this new iteration of the Empire will then include all the other Habsburg possessions, too. (Like the original HRE, it'll inevitably be a decentralised entity, with lots of legally established territorial privileges and tradition-rooted exceptions and local sovereignties and self-governing regions and whatnot.)
On the other hand, the North German Federation will tend towards a more "organised" form, with matters being regularised and standardised. This was a driving impulse of the "modern"-minded liberals, after all.
Soon enough, the two states--Federation and Empire--will settle into a comfortable co-existence. Perhaps not exactly warm, but there will be no reason for conflict. Indeed, the mood may prevail that "this is what should have been done from the start" (e.g. after the Thirty Years' War, instead of the accomodation of Westphalia). A Protestant realm and a Catholic one, clearly distinct, without trying to force it all into one system.
Outside the German world, things will surely proceed apace. Italy will seethe, but will not dare to attack Austria without external support. Napoleon III will not be inclined to give them that support. There is certainly no Franco-Pussian war, and the Germans have their issues sorted out and are at peace. France isn't going to antagonise them for no reason. After all, France is emerging from its post-Napoleonic isolation, but with another fellow named Napoleon on the throne, the French are aware they're on thin ice, a bit. If they start instigating wars, things will go South for them, and right quick!
Britain, meanwhile, maintains its "permanent interests". Liberalising Prussia under a friendly monarch? Good! France not making any trouble? Good! Austria getting its house in order? ...Well, that deserves cautious observation, but no reason for alarm right now.
Which leaves the Eastern Question. The Crimean War has at least blunted Russia's drive into the crumbling Ottoman realm, but it was a bloody conflict, and many consider the price to have been too high. For the time being, Russia won't be willing to openly incur British ire, but the Tsar will give a lot of support to Romanians, Bulgarians, Serbs... even to the Greeks, I'm sure, in an attempt to fully wean them off British protection. Meanwhile, the Habsburgs will have a lot on their plate making sure their restored Holy Roman Empire is actually stable and decently well-functioning, so they won't be serious competitors on the Balkans. As the Turks fall back and their power dwindles, those new Balkan states are going to be nominally independent-- but in practice, they'll be part of an expanding Russian sphere, and everyone will know it.
As such, the long-term outcome here, on the grand scale, may well be that the great conflict of the coming century will be between reasonably united European powers on the one hand (sort of propping up the Turks on the side), and Russia on the other hand (supported by the Balkan states, and possibly with some revanchist players--like Italy--throwing in their lot with the Tsar, too).