Hydrophane leather dressing
is good, but my favorite is Passier Lederbalsam
. It's more natural to leather, restoring it without weakening the leather fibers.
Clean the bridle with glycerine soap
or Lexol cleaner
, then warm some of the lederbalsam up in your microwave (so it's warm, not hot). Then rub it in with your hands. Work it in the leather for a while, then let it set over night. If it soaks all the way in, warm up more and do another treatment. Then you should be able to gently flex, bend, and roll the leather while you put another treatment on the leather. Flexing it will help the conditioner penetrate the leather. Remove the bit and any silver plates/buckles that have a screw back and can be removed, so you can get the conditioner on the leather more thoroughly. Let the bridle set for a few days, laid out on a towel. If there's any conditioner left on the leather, wipe it down with a clean towel. Use a soft old toothbrush to get any out of tooling or other crevices.
Clean the silver like you would your good silverware. Do NOT use TarnX. I like using baking soda. Make it into a thin paste. Use a soft cloth and rub small, gentle circles over the silver. Rinse and repeat until the tarnish is buffed off the silver. Buff with a soft dry cloth to bring out more shine.