No, you use the creature's proficiency bonus on its attacks.

The Wild Shape class feature describes which statistics are kept, and which statistics are changed, you use when you use this ability.

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.

According to this ability, you only you use the druid's proficiency bonus on skills and saving throws. Attack rolls are neither skills nor saving throws, so it does not apply. Otherwise your "game Statistics", such as proficiency bonus on attacks, are replaced.

I also found a relevant tweet by Jeremy Crawford, one of the game's designers:

The intent is that the druid uses the bonus in the beast's stat block for any proficiency the druid lacks.

Unless the druid is proficient with claws, bites, or other types of the creature's attacks, it seems that there is no cause to apply the druid's proficiency bonus.

Remember, the lack of an explicit "no" does not translate to a "yes". If druids could retain their proficiency bonus on attack rolls when wild shaped, then the rules would say so, but that is not the case.

You don't recalculate anything

You simply read the numbers for the bonuses off the beast and PC "character" sheets for skills you and the beast are proficient in and take the higher number.

The relevant rule for your question is:

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. If the creature has any legendary or lair actions, you can’t use them.

Importantly it does not say you retain your "proficiency bonus" (which is the relevant game keyword), but instead says you retain your skill and saving throw proficiencies.

i.e. you retain the modifier for skills and saving throws you are proficient in, unless the beast has a better innate proficiency in the skill.

From the Ability Scores section of the Players Handbook we have the following rule:

A skill represents a specific aspect of an ability score, and an individual's proficiency in a skill demonstrates a focus on that aspect. (A character's starting skill proficiencies are determined at character creation, and a monster's skill proficiencies appear in the monster's stat block.)

Thus the bonus referred to in this rule is the number listed under "skills" in the relevant beasts stat block.

Proficient Skills & Saving Throws

Taking the Ape as an example, it has the following skills listed in its stat block:

Skills Athletics +5, Perception +3

If your Athletics skill on your character sheet is +3, then in your wild shaped form you take the +5 from the Ape.

You would also gain the +3 in Perception from the Ape if your characters Perception skill was less than +3.

Conversely, if you are proficient in Acrobatics and your skill bonus is +3 (assuming a 10 in DEX) while the Beasts is +2 (due to its +2 DEX modifier) then you take your characters +3.

The same holds true for the saving throws you are proficient in.

Other Stats

For every other number on the two character sheets you

• take yours if it's related to CHA, INT or WIS
• take the beasts if it's related to CON, DEX or STR