The answer to your question changes depending on the method of transformation.
The target’s game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains its alignment and personality.
When you are using either of the polymorph spells, you set your character sheet aside completely and play with the new form's stat block in its entirety. That means your character statistics (aside from alignment and personality) have absolutely no effect on anything regarding your polymorphed form. This includes things such as proficiency bonus.
Jeremy Crawford agrees:
While you're under the effect of the polymorph spell, you use the stat block of your new form, not your character sheet.
You're playing with the stat block, not your character sheet. Set the
A target's game statistics are replaced by the statistics of the chosen beast, though the target retains its alignment and Intelligence, Wisdom, and Charisma scores. The target assumes the hit points of its new form...
With animal shapes a character retains alignment, Int, Wis, and Cha scores. All other statistics (including proficiency bonuses) are replaced by the new form.
You do not get to use the proficiency bonus or any other statistics of your original form outside of the listed ones. If it were allowed, it would say so.
Wild shape, being a special druid ability allows your character to retain some aspects of your original form even when transformed.
As for Wild Shape, it specifically states:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature’s bonus instead of yours.
Wild shape is similar to animal shapes but with the addition of also retaining the skill and saving throw proficiencies of your original form.
With regards to proficiency bonus, if both you and the creature have proficiency in something, you take the higher bonus. You never add them both.
Jeremy Crawford has also confirmed this:
The Wild Shape feature does not let you add your proficiency bonus to
the proficiency bonus of your beast form. The first bullet of the
class feature details which proficiency bonus you use (PH, 67).