Prior to the war, France had been vacillating between an alliance with Prussia vs one with Austria. The King's mistress, the Marquise de Pompadour pushed for an alliance with Austria. The Prince of Conti tried to find Louis XV another mistress in 1756. As late as 1756, King Louis XV sent an envoy to Potsdam to meet with Frederick II to see if he could be won over.
If Prussia allies itself with France, than Austria is going to remain a British ally. Meanwhile, Russia and Austria had entered into an agreement by 1756, so it seems likely that these two countries would fight as British allies.
As for the other powers, the United Provinces preferred to remain neutral unless attacked. They were bound by treaty to an alliance with Great Britain, but did not enter the war in OTL. If the Austrian Netherlands are invaded, the Dutch may join the war against France. However, their army was not in the greatest shape by this period.
Sweden too is up for grabs, but it was a declining power by the Seven Years War. It certainly did not seem to be at the top of France's priorities in 1756, so they may just choose to remain neutral. The court seemed to be split about the war, however if things go well for the French, they may seize the opportunity to enter the war late in the game to regain some of Russia's conquests earlier in the century.
Spain had been able to rebuild its navy somewhat by this period, but was not as strong as they would be in during the American Revolution. In a war against Britain, their primary objectives will be retaking Gibraltar and Minorca. Early enough in the war, this could be accomplished, as the British Mediterranean was not well defended. The Spanish preferred neutrality, but with the Austrians on the other side, I can definitely see them assisting Bourbons in Italy by attacking Milan and Tuscany. They may also attempt to retake Sardinia, if the Savoys join the war.
Sardinia had a decent army and they were considered the "Prussia of the South", however like Prussia their army would be reliant on subsidies. In OTL they remained neutral as they were war weary from previous wars. They may be bought by the British to secure Italy against the Bourbons. However, they will rely on the Royal Navy to protect Sardinia.
Denmark remained neutral during the war, but they were in conflict with Russia over the Duchy of Holstein. In OTL this remained an obstacle as France was allied with Russia. In this TL, they may just join the French in return for a handsome subsidy. Denmark had the largest Baltic Navy, and would be valuable in cutting supplies of naval stores to the British.
The Ottomans are another power that is past its prime, but I can see them attacking Russia if the war goes in favour of the Franco-Prussians.
Below is a list of definite alliances in 1756 when war breaks out.
BRITISH
Great Britain
Austria
Russia
Hannover, Brunswick, Hesse, other minor German states
FRENCH
France
Prussia
Saxony
Below is a brief outline of what I can see unfolding if the Franco-Prussian alliance endures.
On the sea, the British navy has the advantage. However, this would most likely make the French press the Spanish and Danish to join their cause. On land, the French and Prussian armies are the two largest and best equipped forces in Europe.
In OTL, the war went well for the French until 1758, and then turned against them. They were able to capture Minorca, most of Hanover and hold back the British in North America (along with their Indian allies) despite the numerical odds being against them. Austria's finances were in poor shape, and its armies were not prepared for war, and they were unable to destroy the Prussians, forcing France to send the bulk of its forces to Germany.
If Prussia is an ally, I can see the Austrian Netherlands and Hanover, Brunswick, and Hesse being overrun by 1757. Austria, had few troops there so it shouldn't be an issue. The question is if the United Provinces feel threatened enough to join the war. The British won't be able to provide further reinforcements as they are already stretched thin (just as in OTL), so this may just make the Dutch stadtholders decide upon neutrality.
With far fewer troops needed in Europe, the French are more than likely able to commit at least another 10,000 to 20,000 troops to French America, which will more than likely hold New France. Not feeling betrayed by France's alliance with Austria, the Spanish are going to join on the French side in 1756, along with Bourbon Italy. This will lead to Tuscany, Milan and Mantua being vulnerable. Gibraltar too may just fall before 1758 in such a scenario (before it is reinforced).
If Gibraltar falls just as Minorca did, then the Savoys will more than likely remain neutral as the British Navy can no longer assist them in the Mediterranean. On the other hand they may just make a play for Milan and Mantua themselves. Having them against Austria will further weaken the Habsburgs.
Russia will try to send troops to defend the United Provinces (at British behest), but this will draw Denmark into the war. With Franco-Prussian successes both Sweden and later the Ottomans are drawn into an alliance against the Russians.
I imagine the war would be over by 1758 or 1759. Prussia may just take Bohemia and Moravia (depending on their success), along with Hanover. France of course would want the Austrian Netherlands, Lorraine and the Habsburg possessions along the Rhine. Denmark could take definitive control of Schleswig and Holstein. Meanwhile Sweden retakes a portion of Ingria, if their successful enough (Livonia), perhaps with Danish naval assistance, they partition it.
In Italy, Tuscany falls to the Bourbons, and Milan and Mantua to either the Bourbons or The Kingdom of Sardinia (depending on if the latter enters the fray). The Ottomans perhaps claim Banat from Hungary and expell the Russian influence from Tauride and Crimea.
Britain may just make peace with France early on and leave the Habsburgs fighting. In OTL they almost did once Hanover fell, after Klosterzeven, but Prussia's victory at Rossbach in November 1757, encouraged the British to keep fighting. In this TL the British would be much worse off on the continent and France would have more troops along its coast to threaten an invasion, along with more troops available for America. This would leave Austria and Russia alone and they too would probably sue for peace.