My bridle is wiped down after every ride, either with a glycerin-type soap or a wipe like Lexol. I’ve yet to use Higher Standards… I’m waiting for my castille soap and glycerin to run through before indulging 
I usually bring my bridle with me to the wash rack and hose off my bit there, while it is attached to the bridle. I dissect my leather weekly, at least - I HATE the black gunk that accumulates in the folds of the reins/bridles in the hook/stud or buckle closures. I’d say I break down my bridle completely at least once a week - I also alternate bridles and on their ‘off weeks’, condition them with an arsenal of things: Leather Therapy Wash, Passier Leaderbalsam, Belharra, you name it. I use Leather Wash fairly often, so my bridle stays sweat/moisture resistant – so I don’t fret too much about when I hose the bit down.
For cleaning the bit, it depends on the bit. If it is something with a porous or plastic surface, I’ll soak it for a while and then throw it in the dishwasher. If it is metal, I’ll soak it in hot water w/ ivory soap, and then scrub and rinse thoroughly. I think the Ivory does a great job of making the bit seem brand new, but I clean my bit so often it doesn’t accumulate gunk much.
But, I’ve used everything for cleaning bits: listerine or denture tabs work great, in a pinch I’ve also used just about any soda on hand: tonic, club soda, coke, ginger-ale, etc - the carbonation does a good job of breaking down gunk, and I rinse it off after! :yes: