I was thinking more along the lines of this happening where France and Prussia are already at war...
I think you are onto something here. As much as it would be in Great Britain's long term interests to protect its Prussian ally, I think they would still be unwilling to get involved. Even at the height of the Seven Year's War GB did no more than raise its subsidies and conduct a few amphibious operations against France when most of Prussia was occupied by foreign powers and Frederick the Great was at his most desperate. Rather than intervene they preferred to subsidize Prussia, and often urged them to accept less favorable terms rather than continue to fight.
I don't think it would be that clear to British politicians around 1790 that Prussia could not handle itself against France and Austria with the support of Russia. Prussia had fended off those two countries forces during the Seven Years' war with Russia on their side. Prussia had only grown stronger (though perhaps more than a bit complacent) while neither France nor Austria was appreciably stronger in 1791 than in 1754. Austria may have actually been relatively weaker, if their performance in the Austro-Turkish War of 1787-91 was any indication.
Russia would probably be more than a match for Austria in the East. 1791 is still the age of Suvorov. That man was the probably the greatest general ever to be born in Russia. He enjoyed the confidence of the Empress Catherine, so he would probably be given everything he needed to succeed. And at that the head of a large army (any army for that matter) he was never defeated in his lifetime. He would march through Poland, defeating everything in his wake, take Cracow, and probably invade Hapsburg Transylvania and Hungary, if not Austria itself. The Russian Army would be more likely to cross the Rhine into France than the Austrian Army would be to even make it into Ukraine.
Even if Prussia was completely defeated, they would not face elimination. At the worst, they would be forced to return Silesia to Austria, and perhaps surrender some bits of Poland that they had taken in the Partition back to Poland. That is all. The King of Prussia would still remain an Elector, the Margrivate of Brandenburg would probably not be touched. Even the scoundrel Max Emmanuel was given back his HRE domains after the War of Spanish Succession ended.
My point is that British policymakers would feel little pressure to intervene in favor of Prussia. The Netherlands would be a completely different matter. GB viewed control of the channel as a matter of national survival. Unlike a potential upset in the balance of power in Europe, this would force GB to take immediate action. However grudgingly, armies would be raised and sent to the Low Countries, as they had been in the days of Marlborough and Cumberland.
Even then, I am unsure that GB would declare war on Austria, unless they decided to declare war first for some insane reason. This would be for the simple reason that they would be nothing to be gained for GB by winning victories against them in the Austrian Netherlands.