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
In general: You chose which AC to use
The answer to your question is very straight forward:
If you have multiple features that give you different ways to calculate your AC, you choose which one to use. (PHB)
Monk/Druid - Can use either
For Monks specifically there is an official ruling by Jeremy Crawford:
The druid/monk could use Unarmored Defense or the beast's AC calculation.
Barbarian/Druid - Can use either
Barbarian unarmored defense is worded exactly the same as the monk's so the previous ruling is applicable here as well.
Draconic Resilience - Beast's AC except, maybe, if the beast has scales
You can't use features that the beast form doesn't have
Per the PHB
can use them if the new form is physically capable of doing so
and Jeremy Crawford
Wild Shape: keep your racial traits, unless told otherwise or they
require external features (wings/hands/etc.) the new form lacks.
You cannot use any racial features that your beast form does not have.
Draconic resilience depends on specific physical features
The Draconic Resilience feature says:
parts of your skin are covered by a thin sheen of dragon-like scales.
When you aren’t wearing armor, your AC equals 13 + your Dexterity
modifier. (PHB)
Note that this effect is predicated on the presence of dragon-like scales on your skin specifically. This means that this feature requires those to be present.
If you wild shape into a beast with skin and fur, the best thus lacks the necessary physical traits to make use of the draconic resilience.
Very similar official ruling
Jeremy Crawford made a ruling about a lizardfolk's natural armor feature. The description of which is:
You have tough, scaly skin. When you aren't wearing armor, your AC is
13 + your Dexterity modifier. (VGtM)
Note how it is very similar to the description of draconic resilience.
His ruling was:
The lizardfolk's Natural Armor specifies anatomy: tough, scaly skin. It doesn't apply if you're wearing a beast's skin in Wild Shape.
However, if the beast has scales it might qualify (DM's choice)
Note that in the above ruling Jeremy Crawford specifically says "wearing a beast's skin". It is possible that if the beast has scales (for example the crocodile or ankylosaurus) that it might qualify as now having the necessary physical feature (scales) thus allowing you to use draconic resilience.
This will likely come down to a DM call as to whether any of the creatures on the list have a "thin sheen of dragon-like scales", but it seems like a reasonable ruling to allow.
Stone Sorcerers - Should be able to use either (if the beast has skin)
Per the discussion above about draconic resilience, if you turn into a beast that has skin you should be able to chose between the Stone’s Durability or the beast's AC.
Note: Stone Sorcerer is UA material so it is non-final.
Best Answer
You must apply the higher stat, no doubling of PC stat
Textual Argument
For a particular skill, there are two proficiency bonuses contemplated in the above text: your humanoid form proficiency bonus, and the beast stat block's proficiency bonus. You propose a third category: a hybrid proficiency bonus with the PC's proficiency as base, augmented by the beast's 2x proficiency feature.
When in doubt, the rules of the game only do what they say they do. Every sentence of this text involves a direction to pick between either a PC number or a Beast number. There is no mention of calculating new numbers. Instead, the text states that you must choose the "bonus" that is higher. Here, "bonus" probably refers to "skill bonus", as defined in MM chapter 8: "A skill bonus is the sum of a monster's relevant ability modifier and its proficiency bonus". The text dictates that you must use the higher of two bonuses: you may not invent a third number.
MM Chapter 8 continues:
The saber-tooth does not have an ability that doubles its proficiency bonus. There is no evidence that "heightened expertise" is anything other than flavorful description. If "heightened expertise" were a real ability, it would appear in the monster's statblack. Again, when in doubt, the rules only do what they say they do. From the MM, all we really know is that a monster has a higher proficiency bonus in a particular skill merely because it says so.
Hit point modifiers are likewise determined arbitrarily. Bugbears and Copper Dragon Wyrmlings are the same CR, with the same Constitution, but the Wyrmling has +4 and the Bugbear has +5. That's just how it is.
Because the higher proficiency bonus is not tied to an ability, you have no basis on which to apply said nonexistant ability to a proficiency bonus imported from your druid.
Purpose Argument
The 5e Bounded Accuracy system is meant to reduce the amount of math that a player has to do on the fly. This is in contrast to Pathfinder, where a druids may resort to excel spreadsheets to calculate how different modifiers and buffs apply for all of their various beast shapes.
DMG Chapter 7's section on Proficiency Bonus does not speak explicitly to your question, but it emphasizes the focus on picking a single number and applying it once:
Note that, in the above text, a doubled proficiency bonus is not treated as a proficiency bonus applied twice: the doubled proficiency bonus is the new proficiency bonus.