Clerics and inquisitors are both proficient with their deity's favored weapon ... what happens if they decide to worship a different God?
Do they auto gain a new weapon Prof and lose the previous?
Yes, but I'd say worshipping a different god wouldn't be an easy task. They would have to break bonds with their current god and form bonds with the new one. I'd allow the proficiency change as part of this process.
You would have to ask your GM, but there are no rules for changing your deity. You could argue it is not possible within the rules.
I'd probably have an atonement spell cast by a priest of the faith and require some retraining, a sacrifice, donation or quest might well be part of it, at least enough to pay for an atonement spell.