^ these answers I've given should be pertinent.
Short answer is that it didn't. There's no way that a movement that was so small, and which seems to only have affected Britain, could have made any measurable impact on the outcome of the First World War. More importantly, it's impact in Britain is highly exaggerated. Of the almost 5 million Britons that served in the British Army in WWI, over half were volunteers, approximately half a million were already in the armed forces in the Regular Army and Territorial Army, and only c. 800 000 were recruited by compulsion, from spring 1915 until the passing of Conscription in January 1916 (the rest of that 5 million were conscripts). Those c. 800 000 recruits from compulsion were almost entirely the product of the Derby Plan, whereby recruitment officers visited men eligible for service that had not enlisted and either convinced them to voluntarily enlist at the moment, or to sign an agreement to enlist at that particular recruitment station in a couple months (c. 6 months) time.
White Feather giving seems to have been sporadic in 1914-15, and petered out by 1916 with conscription, with only a few scattered cases in 1917. It does not appear to have been very popular, and most men particular among the Kitchener Volunteers do not appear to have required overt compulsion to enlist, given that 1.5 million had volunteered by December 1914. If you're interested, my previous answers have sources and videos which address this in particular.