I'd have to disagree here. I think it's arbitrary to cast philosophy and science as opposing, let alone as others. Science, to me, is just an extension of modern philosophy - one that is more rational and reality-oriented. I don't see how philosophy and science can coexist as you explain it. If we go back and convince Aristotle that his a-priori knowledge too are founded on assumptions that really have no bearing on reality as we see it, I'm thinking he'd come around to abandoning his ideas as insignificant speech. Or maybe we couldn't if he's too deep in his constructions to escape the cave :P would he be able to see the irony there (it's Plato's work, but still)?
But whatever, let's say he's able to see past "truth" and throws all that away, now thinking of the world as matter in motion. His classical philosophy of old would still disappear. Aristotle was trying to explain reality, but that reality has become obsolete. The assumptions would not stand. I suppose his philosophy could guide morals, but perhaps Aristotle would even see through his assumptions on human nature. Even come to the conclusion that there is no truth, or at least humbled enough to admit that there is more for him to think about.
Even with an accelerated development of science, its fruits would not reach the masses until an industrial revolution. And such a circumstance needs to reward constant (or as much as it can be) innovation. I'm talking about capitalism and the profit motive, the pursuit of using machines to make more, better, quickly. Without a concentration of resources and mass production, the lifestyle we have today could not come about. Even with the development of basic chemistry and optics pre-industrial revolution, people largely lived the same way the did for hundreds of years, producing food and manufacturing on a small scale. But that's a digression.
We have come from believing there to be a truth to undermining what we thought to be truths to discovering reality, just once again to undermine what we thought to be reality to discover how we construct reality. I don't think philosophy and science coexisting as you've described it to be possible :\ Each step, from the classics to modernism and now post-modernism shakes up the old paradigm, makes it look parochial - too small for an ever bigger world with ever bigger issues. In any case, fret not for we may be soulless, but we are still philosophic as ever!
I do think recognizing the scientific method several centuries ahead of time could very well accelerate technological innovations. I think it's pretty reasonable to suggest that some of our major scientific breakthroughs could be bumped up a few decades, maybe even a few centuries.
To go back to my example, what if Hitler were assassinated before he took power, I think with Hitler & a re-militarized Germany out of the way, Europe would become either new Roman colonies or extensively Russian. The biggest check to Soviet expansion is out of the way - nothing in eastern or central Europe would be able to stop Stalin. Add Mussolini to the mix, and oddly enough I think we have a pretty similar scenario, with the French & English and the other western European nations resisting a fascist dictator in Mussolini. And a massive, probably aggressive, Soviet nation under Stalin, who was completely ruthless. This all assuming Soviet Russia and Mussolini get along, which makes some sense given that originally, Nazi Germany & Stalin's Russia had a non-aggression pact - Soviet Russia invaded Poland and some of the Baltic areas also.
So we still have an axis, just with a perhaps even more fearsome enemy in Russia, because they didn't let a fanatical madman make military decisions. Part of Nazi Germany's fall was that Hitler ignored the advice of his top brass and made strategically poor military decisions that cost the Reich. So instead of that, you have Russia with its harsh climate, giant amount of territory and resources, and a disciplined and formidable army. So we're essentially in the same boat, minus that the Holocaust & accompanying racial cleansing may not have happened. That's well over six million less casualties.
I can't say that a war with Imperial Japan, facist Italy, and Soviet Russia would have gone any better for the allies, it may very well be worse. But, the United States still has the bomb in 1944-45, sooo.
Or does it? Albert Einstein penned a letter to FDR about the future of atomic power and how its destructive power could be harnessed for military means, and that the Germans were going to pursue it, with Germany having been an intellectual center of Europe at the time leading up the World Wars. Germany supplied many of the minds working towards the atomic bombs/harnessing atomic power. So without that Germany, there may not have been a bomb to race to perfect. Just like there wouldn't be a U.S. space program without the Nazis. (Little known fun fact btw) So, we may not have had a nuclear bomb to use as a last resort to win the war.
As a student of realism, I would imagine that Germany would still re-militarize. It still has its industrial capacity and its manpower and its economic power and its key central location in Europe. Any statesman would look at that picture and say "something ought to be done with that potential". And the German people would still be upset and nationalist over the treaty of Versailles. Germany would not see itself contained by France and Poland, especially when it has the resources to break out and regain its preeminent position on the continent. Britain and France, and especially the United States, would be more than happy to appease a more rational and less aggressive Germany. I could see them annexing the Sudetenland in the name of German nationalism, but keeping the rump Czechoslovakia as a satellite, perhaps broken. Austria would follow sooner or later. I could see them getting away with a limited war on Poland, with a limited scope of reclaiming the old borders. Even if France and Britain declare war on Germany, she will accomplish her war goals long before either of them can help and present the re-annexation of her lost territories as a fait accompli. The war would probably fizzle out, as Germany would not be threatening any other countries and it would be better to cut their losses with a slightly smaller Poland than another European war. This Germany's foreign policy would be more schrewed and less aggressive, arguing on behalf of its security and unequal treaties.
At this point the situation in Europe is highly volatile and anything can happen. There would probably be an uneasy peace as German nationalism and irredentism is fulfilled. Will there be a Soviet threat? How could a war start? What if Germany is losing? Would Britain or France intervene? :P
On the other side the planet, Japan would probably carry on its invasion of China and establishing its sphere of influence in Southeast Asia. Sanctions against Japan would occur, probably a war in the Pacific. Again, anything can happen. Would the Communists win? Could the Nationalist Party receive support from both the US and Germany? Due to the military parity, could there be a coalition between the Communists and the Kuomintang? <3 Or would the civil war just be slogged out for longer?
TL;DR - Alternate history is fun!