I used to work with leather.. (making tack, bags, dog collars etc).
DO NOT use saddle soap, saddle soap is neither a soap nor actually a cleaner, its more a moisturiser, its a humectant that draws moisture in and keeps leather supple and actually it isn't all that good for even bridle leather, but bridle leather is so tough and strong it survives despite saddle soap rather than because of it.
Notably i have things cleaned with saddle soap and things NEVER touched with it, in my damp workshop... guess which items have moulded?! Yep, only teh saddle soaped stuff.
Clean it very gently with lukewarm water, use a TINY miniscule drop of washing up liquid if you think you really have to do so.
Dry it with a cloth and immediately (whilst its actually still slightly damp) polish and buff with a leather conditioner made from FAT and WAX (not oil)... this will feed the leather and seal IN the moisture from the wipe down. This will be a solid product like Abbey Saddler Leather Balm or Jeffries leather conditioner and you need the warmth from your hands to work it into the leather, then let it sit and buff it off.
Avoid oiling leather, oil drives out moisture and allows the fibres within the leather to soften and move, eventually resulting in rather crispy leather or 'soggy' feeling soft baggy leather.
Saddle soap pretty much does the opposite. Neither are actually good for leather.