I agree with most of what was suggested...
Clean well as this will open leather pores to accept nourishing nutrients in the cleansing soap and oil.
Just cleaning with quality saddle soap or glycerine bar will make a big difference in pliability of the saddle.
Water on leather will not hurt it as was said.
I soaked my western saddle fenders from the un-fnished un-tooled side when we needed to turn them, placed a broomstick to hold the twist and let the leather air dry in the shade...repeat several times till when you release the broomstick the stirrups maintained the look desired...
Do the same idea to those skirts to flatten them down...you will probably need to soak where you can so front, outside seen is where you will also need to apply
Pad the skirts and strap them down in place, or weight the leather to get back the shape you desire as the leather finishes drying and learning its new position to hold.
Apply several coats of pure neatsfoot oil but do not soak the leather...apply light coats and let soak in between applications.
After oiling and the leather no longer absorbs, wipe the excess off the leather gently...
Buff with a soft cloth for added luster to come out and shine with health restored.
If the seat is suede do not oil it but use a short bristled brass brush or a suede brush found in some shoe departments to raise the saddle seat nap...if stitched be cautious you not pull the stitching loose or break them.
Then you need to lay the leather down again, pad and secure it in the shape it must retain...
To secure I've seen soft nylon ratcheting straps applied after blanket/pad were put in place, then straps and tighten so it is held in place.
I was told not to put in sun but let water or oil soak in in shaded atmosphere but warm conditions like summer warmth/heat is fine.
Rapid evaporation is not the best for leather...slow, steady absorption was the way to proceed.
Never tried this in the sun, ever...always in shaded area.
Do not soak the leather with excessive oil...old stitching can break very easily when over-oiled so be very careful.. hence several light layers applied not saturate the leather may be the safer way on a old saddle to proceed.
Good luck and pictures of that finished beauty when project completed...even dry the leather has a quality patina to it. WELCOME to the Forum!!