Once I was showing and it started to rain really hard so my saddle got "bleached" (is a calf skinny CWD so very sensitive). All I did was keep conditioning it. This would probably work for all leather. In general just wear rain coats and a rain sheet for your horse! Obviously don't go outside if it is raining super hard
Rain sheet for your horse, rain over trousers for you, and a rain jacker
Waterproof clothing is the key to being able to weather the rain, it's pretty basic, but it's the only thing that works. Either you go a jacket to climb that is waterproof, or you put a simple raincoat over it (there are transparencies for the competitions). Afterwards, for other tricks: to provide a change of clothes, to lubricate your shoes with a special grease for waterproofing and against moisture, to put the horses under the bomb or in the jacket, to provide a cover for the saddle and to avoid white pants.
Have a good day,