Wow Guys,
Thanks again. What a can of worms.
The original Winchester stocks have a distinctive red hue, which is difficult to reproduce, so for that reason I would not try the Old English brown colored cleaner. If it hits any wood where the finish is worn through, it will dye the wood brown.
One thing I have used on old furniture is Old English pure Lemon Oil. It is a very light oil, and its citrus really does cut through grime. The oil that hits bare wood does not seem to change the color, and it is so thin that it seems to evaporate in a few days.
Unfortunately, the current patina/grime of the stock is sticky, and leaves your hands stinky, so it has to be cleaned.
I think I will go with lemon oil first, very sparingly, and then go upwards to more powerful things in stages, if necessary, from there.
Best, Mannyrock