THis is going to be hugely deperndent on exactly how it comes about, I'm afraid. German politics was complicated. However, I think there are a few things that can be said.
- Relations with the Habsburg Empire - if the Republicans get their way, there will be no Habsburg Empire. The fact that one still exists means that the Republic is to some degree or other kleindeutsch. That will make things difficult in all kinds of ways. With Austria integrated into the German Republic, on the other hand, Hungary is not likely to stay Habsburg or an Empire.
IOTL it was mainly Russian troops that kept the Habsbnurgs in power. Do you see them intervening ITTL? in that case they'll most likely prop up both Austria and Prussia (the Holy League), which suggests the German Republic ultimately will be a small rump state (can it include the Rhine Province? I think not, but who knows). It will need a protector. I'd say France.
If Prussia is part of the republic, but Austria is not, it will be powerful enough to stand on its own two feet, and relations with Austria are bound to be fraught. Of course Austria's in a bad position in the long run here, but in the short term that doesn't matter that much. It might even come to an accommodation with Russia on the basis of imperial hegemony and be viewed as an 'outpost of Tartarism in Europe' or somesuch.
Relations with France are bound to be better that IOTL because worse is hard to imagine. That doesn't mean good, though. A German Republic created in 1848 is going top immediately grow all kinds of irredentist claims, and with its centre in the southwest, Alsace-Lorraine will be the most acutely felt. A weak state will have little choice but to become a French or British satellite, more likely French, but resentment will rankle. A strong one is going to define itself in no small measure against France. If there is no war, this tension may eventually cool off and lead to genuinely cordial relations, but not immediately.
The structure will be federal. There is no other way. German political tradition throughout the modern era was federal. A centralised state would not be accepted, even if imposed by the bayonet (and a Republican state can't really do that). The likeliest outcome will be a remodelled and upgraded Bund with a meaningful army, enhanced legislative powers, and a federal head of state.
Foreign policy could be almost anything. A republic that feels strong is liable to start pushing around its neighbours (Denmark over Schleswig-Holstein, potentially Switzerland over joining, Belgium and the Netherlands over German minorioties, France if it's feeling suicidal, and whatever the eastern border looks like, potentially anyone with a German minority there, too). German Republicanism was strongly nationalistic, and Germans were a very widely spread out ethnicity with plenty of options for conflict. It might also decide earlier than OTL that colonies were a necessity to settle its surplus population (the liberals loved the navy, and many had colonial ambitions). It's unlikely to ever get anywhere, but the implications for relations with Britain could be worrying. At the same time, without the Prussian government forcing a strong army, the parliamentary appropriations are more likely to focus on defensive capabilities, meaning that the actual warfighting capacity will be limited. I don't see the German Republic ever playing in the same league militarily as Wilhelmine Germany.