I have had leather saddles for over 30yrs. I use Neatsfoot Oil, the PURE stuff, not the 'compound', available at most farm supply stores, and, of course, tack shops.
If the saddle needs any reshaping you best to it before you oil. Figure out what you need to do and have prefit forms or string wrap at hand, then wet the leather and form it to shape and set to dry (slowly; it can take several days). Often the skirts are the most mis-shapen and as the leather shrank they flare out so you want to draw them in.
Repeat if needed.
When in shape and dry set it in the sun to warm and then apply the Neatsfoot oil generously top and underside (if you use a hair dryer use the LOW setting, if it is too hot to hold your hand in front of it is too hot for leather; leather, after all, is.....) NEVER use an oven, even on low!!! Baked saddle is not good :shock:
Work a few square inches at a time, returning to those areas that soak it in fast.
First 2 or 3 days to it morning and eve.
Next 2 or 3 days do it once a day.
Next 2 or 3 weeks once a week, or until you can see/feel the leather has become supple. You do not want it to be soaked, but have oil in the leather fibres. You know what good, healthy leather look and feels like, we all have some around. I like to buff in a mostly beeswax top coat for a final (beware wearing light colored pants until youve 'butt-polished' it out good).
The big problem with a 'hamock' style saddle, such as Brooks, is that all the rider weight is really suspended between the nose piece and rear support. Dry rot can make the leather too weak, even if oiled. If it is 'crumbly' it is done. After oiling, if the leather at the nose and the rivets seems to be giving way, it is a gonner but for display purposes. No sense in totally wrecking it if you see it is starting to go.
BTW, The steel pan saddles are a different story, the pan is supporting your weight so the leather can be in too bad a condition to be a hamock style, yet last pretty well.