If the leather is tanned via oak (bark tan aka vegtable tanned) which most western saddles are then there is a wide variety of applications one can use. PURE neatsfoot is one product I use often. Neatsooft oil is from the boiling down of cattle bones. The oil from carteliage and connective tissue rises to the surface in the boiling process and is super filtered and no additives are used thus being "pure". 'From the cow back to the cow' is what I say. Neatsfoot compound has additives that can eat out natural fibers like cotton stitching. NEVER use petroleum products on leather, you will damage the leather (and stitching, esp nylon based threads). Olive oil, I don't use simply because over time it will smell esp if left in the sun to long (which you should not do to leather goods anyways) and if wet. Its a funk you will not get out, some don't mind it but it makes me gag. It might also cause some saddles with cheaper tanned leather to bloch. You do not have to "condition" leather to get it to absorb oil but it must be clean. Do not wet a saddle before oiling water and oil do not mix and the point is moot. Dry clean leather is best. Leather must be able to breath and clogging the pores can damage the leather. If you want a "thinner" oil/conditioner use Leather New Replenisher/Restorer. This absorbs very nicely and I use it often on older leather.
I will warm up neatsfoot oil by placing the bottle (i buy in gallons because I detail and repair them and its cheaper that way. And put them in pint size bottles, easier to work with) and set them in hot water, don't put in microwave....if you get it to hot it can scrorch the oil and thus break down the properties of the oil and ruin the leather. I apply in layers allowing each layer to penetrate the leather. With areas of a saddle I want pliable I will work the leather with my hands while applying the oil. I use a paint brush to apply the leather but you can use a small sponge or rag. (I don't use much in sponges because they hold bacteria and mold. However if you designate a sponge just for oiling on clean saddles you can get away with this. Clean out sponge with dish washing liquid like dawn after applying the oil and let dry, that way bacteria and mold wont get in and party. Use that sponge only for oiling.) I often use my hands to oil strap goods and will rub it in my hands to warm it up some and have at it working the leather while applying the oil. With hands and paint brushes you waste less of the oil. Use a tad of glycerin or leather finisher to "seal" in the moisture and buff to a nice matte finish.
If the oil smells funny (other than its normal oilly smell) and is cloudy do not use it, it may have gone rancid......yes neats foot oil can go bad, remember where it comes from. However in right conditions it has a long shelf life.
Note: Neatsfoot oil wll darken lighter coloured leather.
After a good cleaning and oiling: use a light conditioner between cleanings to keep leather from getting to "thirsty" and to keep it rich.