It is really important to immediately oil it!
Depending on the type of leather (open grain, full calf, leather with a protective sheen) I wouldn't oil and then condition lightly. Let it dry before doing anything else
I like to use tack waterproofing products (there's hundreds). Right after something gets rained on or too wet. Once it's dry oiling it is what I find best
There are a water-resistant leather conditioners out there. I believe Leather New is one of them, I use it and I'm really satisfied with it :)
There is also tack cleaner that makes your tack water resistant. I have used it. It's like a leather conditioner that helps when it gets wet. I don't really know how well it works, I just used it once because I ran out of other tack cleaner.