My understanding is that comercial "oil" finishes on walnut stocks are a penetrating dye for uniform color, followed by a light poly seal coat. They are intended to give the appearance of a hand rubbed oil finish, but using a production compatible process. There is usually little to no buildup on the surface of the wood, but it is sealed against moisture.
The Minwax Antique Oil is made of linseed oil plus driers, the Royal London is probably the same type of product. These are rub on, rub off products where you let the finish soak in then rub off the excess. The Miwax starts to dry in minutes, so you can't wait too long between application and removal of any excess. The Royal London might dry a little slower and be easier to work with. You don't want buildup in the fishscales. I have used an old toothbrush in those areas to remove any excess before it dries.
If someone has used the Royal London to add depth and sheen to an AA stock and had good luck, I would just do that. To get started you might cut the finish with about one third mineral spirits to improve penetration and slow down the drying time. That's what I do when finishing furniture to control the process and slowly build to the desired sheen.
Good luck!