You recalculate skills and saving throws a proficiency bonus is applied to:
- Skill and saving throws based on Str, Dex and Con in which you have proficiency
- Skills and saving throws based on Int, Wis and Cha in which the target beast has proficiency (not common, but perception is the usual case)
You should "extract" the beast's standard proficiency bonus by subtracting the total attack modifier or skill modifier from it's related stat bonus. Example:
The beast has a Strength score of 16, and thus a +3 Str bonus. If it's melee attack modifier is +7, then it's proficiency bonus is 7-3= 4.
Then, If your proficiency bonus is greater than the one extracted, you should use yours to recalculate the skill, saving throws.
As a DM I tell my druid player to keep a record of the adjusted stat block of the 3 most used beasts, and adjust them when his proficiency bonus changes, to speed up the gameplay.
According to Jeremy Crawford, the druid has no proficiency with natural attacks, therefore the attack bonuses should not be recalculated