I've got a couple of wool day / travel rugs that have leather breast straps and I've always just oiled them well before & after washing, not had a problem yet but guess the alternative is to take them to be professionally cleaned.
Wash on gentle wool cycle but be prepared for the wet/damp leather straps to mark the rug - hand washing is possible but they get very heavy to handle. Shrinkage is the main problem I found - mind you that could be due to washing one at too high a temperature and then putting it in the tumble dryer I have now decided against wool rugs and use ones that can be hot washed at home.
I greased the leather really well then wrapped a bit of cloth around the buckle to save my washer, then put sandwich bags round and round the leather bits as many times as they'd go round and fastened them with hairbands. I figured some water would get in from the back but I could minimise it at least.
Then I used a conditioner on the leather afterwards. The rug was about 5 shades lighter than I remembered, lol.
Ive messaged the manufacturers also. It says on the website the rugs are previously washed before they are cut to minimise shrinking. Its a Witney rug.