The plain blue Dawn diluted with warm water these days is about the best choice. You can use a hose and rinse really well use a soft brush on the tooled side to get the gunk out if the carving, stiffer brush on the underside on bad spots. Western saddles are far more awkward to really clean then the little English ones you can wipe down in the comfort of your living room watching TV. I’ve used a wash rack as well as my patio garden hose. Let dry at room temp out of the sun and away from heat source.
Warm your choice of oil, flip the thing upside down, spread the fenders like wings and use a 1" -2" width paintbrush to paint the underside of everything and up under all the nooks and crannies. Let dry. Do the underside again. Let dry. Multiple light oil coats allowed to dry and soak in between application, not a thick, single coat. The conditioner products don’t soak in as easily and quickly as plain old saddle oil.
Do the back of the fenders too, confditioning tooled leather on the untooled or rough side is often ignored but that’s where the dirt, oils and horse sweat can get imbedded and compromise the leather over time.
When it stops sucking up the oil like a camel coming out of the Sahara to an oasis pond, you are where you should be. There are a number of good products out there for the topside of the saddle including dyes and stains but most don’t mix well with the straight oil you’ll be using on the underside and the " shiny side" wont usually absorb the plain oil very well and can get blotchy if you apply a stain over where you oiled. Probably want to give it the same really good cleaning at a regular intervals, I did and still do it about every year if the saddle is in regular use. So do my friends with their custom Reining and Cow Horse saddles from today’s most respected makers.
I had a saddle maker out in So Cal named Forest Knott make my show saddle back circa 1970 and he thoroughly trained me in its proper care before he let me take it. My saddles have always lasted loooong time and resold well following this, English or Westen. Back then it was Ivory, Castile or Murphys Oil soap but you can’t find those easily today or the formula has changed. There was nothing special about those, just soap with no additives that didn’t leave a residue, plain old diluted blue Dawn is the closest you can get today and, really, if they use it to get crude oil off ducklings ( which they really do) it’s OK for your saddle and it will rinse clean as products with additives do not. Can’t beat the price either.