Let's just for fun take the other side--Prussia stuns and routs the Austrians in Bohemia and repels the Russians in East Prussia. Is it so impossible? Austria is a basket case and Russia is overrated (as everyone finds out just a few years later).
So as not to push their luck, the Prussians take a quick peace. The North German states join the Erfurt Union, now called the North German Confederation. The South is left in flux, but they remain associated economically to Prussia.
Austria is in disarray. There will probably not be renewed full on revolts in Hungary and Italy (too soon), but things are a mess. Schwarzenberg is out and there will have to be changes. Some kind of conservative federalism is the first thought that comes to mind.
Russia experiences a loss of confidence and looks inward.
The center of future conflict now shifts from the east (no Crimean War) to Germany and Italy. Prussia will push for a favorable settlement in Schleswig-Holstein. France will have its eye on Italy with a weakened and isolated Austria, but will also be leery of a strong Prussia across the border.
Do Austra and Prussia reconcile? Is the deal Austria accepts the new arrangement in Germany in exchange for support in Italy and the Balkans?
How do matters progress?
Your scenario is unlikely. The Prussian military was relatively weak in the 1840ies and 50ies compared to the 1860ies and 70ies. While Prussia had some advantages in 1866 against Austria, the Prussian victory wasn't inevitable but the result of good planning and a lot of good luck. If you fight against Austria and Russia at the same time, you need to further split and divide your armies. You want to defend Königsberg against the Russians. You need an army against medium sized German states like Bavaria and Saxony, to defend the RHineland and connect it with main Prussia. Your main forces should concentrate against Austria.
Such a war could also include new violence in Schleswig-Holstein. If Prussia defeats Denmark again, Great Britain will maybe intervene on the side of Denmark.
In this scenario you could gain support from German nationalists and various smaller German states could join your case (but Bavaria, Saxony and Kurhessen (Hesse-Kassel) are probably against you).
I guess that Hannover will be either neutral or supportive of Austria (depending also on the British position). Great Britain could either mediate or maintain neutrality or support Austria and Russia.
But the support for Prussia among some nationalists would be limited, since Prussian military recently restored the monarchy in the Grad Duchy of Baden trough a military intervention and a short war against the revolutionary government of Baden. We need to know how many Prussian troops are still in the Grand-Duchy of Baden in 1850 to explore this possible war.
In most cases Prussia will lose this war. If we want to look at possible peace treaties we need to think about a treaty which maintains the balance of power in Europe while weakening Prussia. The map posted by
@Gloss is one of several possible results. Great Britain will probably influence the peace conference with the interest of maintaining the balance of power.
I doubt that Austria could annex more than half of Silesia. Creative solutions for the Rhineland would be possible, but a Prussian presence in the Rhineland is useful to keep french influence outside of the German Confederacy. Most territorial changes would probably strengthen medium sized countries.
- give Saxony some or all territory lost in 1815 back.
- Bavaria could gain some territory around the lower Rhineland and near the Bavarian palatinate
- Denmark will have a much more powerful position in Schleswig-Holstein
- Minor territorial exchanges between various smaller states.
- Mecklenburg-Schwerin could maybe gain Vorpommern (if they are on the right side)
- Oldenburg could loose some territories to Hannover (if they are on the wrong side)
- Hesse-Kassel could gain or loose territories, but strengthening a reactionary Hesse-Kassel would be in the interest of Austria and Russia.
- Maybe giving Württemberg or Baden control over Hohenzollern (if they are on the right side)
- Russia could maybe gain some areas in the province of Posen (but not to much since we need to maintain the balance of power)
- Austria could gain the southern half of Silesia (but not to much since we need to maintain the balance of power)
The war could strengthen the relationship between Austria and Russia, therefore leading to some changes in the further history of the 19. century.
German Unification is now either postponed or lead by Austria into an reformed closer German Confederacy or will occur if Austria collapses in the 19. or 20. century.