I've got Stubben, Thorowgood, Calcutt &Sons, Freemax and I love them all. You may as well clean a wet saddle, just a wipe over with warm water and soft cloth, since its already wet. There's no point putting leather conditioner or oil etc on top of dirt and seems a bit pointless to dry it out to then get it damp again.
ideal, pessoa, gfs, amerigo, prestige etc
Personally id clean with warm water, allow to dry then oil or use a conditioner and waterproof again with a good leather soap or balsam.