It's tough to find original technical data on stock manufacturing and finishing for LE stocks, but there are military maintenance manuals that stipulate that stocks will be maintained with RAW linseed oil as required. The same directions applied to stock maintenance on various US made rifles incl the M1903 Springfield, M1 Garand and M1 Carbine. We were also provided with tins of RAW linseed oil to re-condition FN rifle stocks when required.
At time of manufacture M1 Garand stocks were treated by a 5 minute immersion in pure tung oil and then allowed to dry. I have a copy of a 1916 dated technical data package on manufacturing the M1903 Springfield which stipulates that finished stocks and handguards will be immersed in BOILED linseed oil and then allowed to dry for a 24 hr period. Presumably, but not stated, these oils would have been heated to facilitate penetration of the wood. It is reasonable to assume that similar measures would have been taken to treat LE stocks, given the volume of production and the urgency of wartime needs. There would have been no time for piece by piece hand staining or finishing of stocks.
I have never seen any information that stocks were to be stained at time of manufacture. There are a lot of variations in military stock color which is a function of variations in the wood used as well as different types of wood. Walnut was the preferred stock material for LE rifles, but Beechwood and Birch were used as substitutes. Changes in stock color are due to a number of factors including oxidization of linseed and petroleum based oils in the wood and the effects of sunlight, rain, sweat, dirt and miscellaneous grime over time. The net effect of all of this would have been a gradual darkening of the wood over time. You can see the same darkening effect in hardwood flooring.
I've installed quite a few NOS Garand and M1903 stocks, as well as some NOS NO4 LE stocks, and prefer to treat them by rubbing in several applications of pure tung oil making sure that any excess oil is wiped off the surface of the wood so that it does not accumulate and dry on top of the wood. The last thing you want in a military rifle stock or any other piece of field gear is a shiny finish which would disclose your position. You can get a nice pimp shined stock by repetitive applications of hand rubbed BOILED linseed oil, but this is not what the military did. Linseed oil was used for a long time based on it's ready availability (extracted from the flax plant which was grown all over Europe and N. America) as well as its ease of application and preservative qualities. The reason why the US Army went to tung oil, rather than linseed, when finishing Garand stocks was because they felt that it was less liable to smoke and sweat when hot. There are reports of Garands being fired rapidly and for such prolonged periods in Korea that the handguards would actually smoke and become charred from the intense barrel heat.