I know from a well voted for answer in Is a Wild Shape attack considered an unarmed strike? that it is possible for a druid to make an unarmed strike while wildshaped, just instead of the beast's normal damage you would deal 1+STR (or otherwise for a monk or druid with Tavern Brawler), but what would the attack modifier be? A character is proficient in their unarmed strikes, so would they use proficiency bonus + strength modifier? Or would it just be straight strength modifier because they have no proficiency bonus, as their game statistics were replaced by the beast's?
You can add the beast's proficiency bonus
The Player's Handbook section on combat says of melee attacks1:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
It also says (PHB, 189):
This chapter provides the rules you need for your characters and monsters to engage in combat, whether it is a brief skirmish or an extended conflict in a dungeon or on a field of battle. Throughout this chapter, the rules address you, the player or Dungeon Master.
So the rule that grants your player proficiency with unarmed strikes also grants the beast proficiency with unarmed strikes.
However, the description of Wild Shape says (PHB, 67):
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.2
This explicitly does not apply to weapon or tool proficiencies. You only have a weapon or tool proficiency if the beast has that proficiency or if you gain that proficiency because of a "feature from your class, race, or other source"
The PHB says (p. 11):
Your character receives a number of benefits from your choice of class. Many of these benefits are class features—capabilities (including spellcasting) that set your character apart from members of other classes. You also gain a number of proficiencies: armor, weapons, skills, saving throws, and sometimes tools.
So proficiencies gained normally from your class (eg. simple weapon proficiency) are not considered a class feature. I do not think the general rules of combat can be considered a feature either.
As such your character cannot use their own proficiency bonus for unarmed strikes but must use that of the beast. This can be fairly easily calculated by looking at the beast's attack bonus for their other attacks and subtracting the appropriate ability modifier. If that doesn't work you can look at the beasts skill proficiencies or consult your DM for the rules about calculating the proficiency bonus for homebrew monsters.
1: This was changed in 2015. Before unarmed strikes were listed as a simple weapon and so you had proficiency because of class proficiency in simple weapons.
2: I think it is clear that this second sentence only applies to proficiencies that are retained not to any shared proficiencies. I am open, however, to the reverse, in which case you could add either your character's or the beast's proficiency bonus.