I have been involved in fine wood working for many years,and in my experience, unvarnished fruit wood will benefit from an occasional light application of oil to prevent it from drying out.
Linseed oil is a drying oil and it is not for this purpose.
It is used as a component of varnish, or is used by itself (more likely "boiled linseed oil").Linseed oil drys very slowly and your bow will be sticky for along time!
Instead, use lemon oil or even lemon pledge furniture polish.(contains lemon oil)
In the '40s, they used phenolic resin for glue so lemon oil won't effect it at all.
My 2 cents worth hope it helps