You would probably still see recognition of the CSA simply because there lies a collection of states not swearing allegiance to Washington, so why have a state of war with them?
You wouldn't have a state of war with the Confederacy, there would just be neutrality. The question then becomes, "Why would Britain and France not recognise the Confederacy?" Well, for a start, for the same reason they didn't recognise the Taiping rebellion during the Second China War: why risk being on the wrong side of history by backing one side before the outcome is clear? For another thing- as much as it's overlooked on here- there's the slave question.
“England has too cordial a detestation of slavery to unite even in appearance with the South." (Leeds Mercury, 30 November 1861)
“There is not- and it is a proud boast- a single paper in the United Kingdom that, following the example of the debased New York papers, proposed to make the most of the difficulty of America, for former affronts to this country, by recognising the Southern Confederacy in retaliation for the act of Captain Wilkes.” (Bradford Observer, 5 December 1861)
"There can be no peculiar sympathy between England and the new Confederacy so long as slavery is the base of Southern institutions" (Saturday Review, December 1861)
"The gulf placed between us, between the English nation and any community that subsists by slavery, that makes it impossible that we can ally ourselves either by sympathy, community of feeling, or affection with any nation that exists upon that which I can call nothing but a disgrace and curse to mankind (Cheers). (Edward Horsman, MP, December 1861)
"We pray earnestly that God may avert from us the great calamity of finding ourselves striking hands with Southern slavery" (Eclectic Review, January 1862)
"England allied with a Confederation of slaveholding states... such a position as this would have been a cruel necessity... Thank God, we have escaped the danger." (Illustrated London News, January 1862)
True, but remember that the South has less foreign currency than the North. Indeed, given the decision to hold onto rather than sell its cotton, it might have run out of what little it had in fairly short order. They had already lost the bidding war on the large number of Enfield rifles being carried by Union soldiers: what weapons were left in the European market by early 1862 were already those from the bottom of the barrel.Even without direct military aid, Britain would have helped the CSA indirectly by making it impossible for the Union to blockade the CSA. Also incoming trade would have brought in more materials with which the CSA could have continued its war with the Union.