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Teferi, Hero of Dominaria's emblem says:

You get an emblem with "Whenever you draw a card, exile target permanent an opponent controls."

Progenitus and True-Name Nemesis both have protection from "everything" opponent controls, which would presumably include planeswalker emblems. However emblems are unusual things - they're not permanents and survive a "Destroy all permanents" effect, have no colour even though the planeswalker creating them can have colors, etc, so I'm asking this question to make sure.

You cannot exile Progenitus or True-Name Nemesis.

"Protection from everything" has a specific meaning in the rules:

702.16j “Protection from everything” is a variant of the protection ability. A permanent with protection from everything has protection from each object regardless of that object’s characteristic values. Such a permanent can’t be targeted by spells or abilities, enchanted by Auras, equipped by Equipment, fortified by Fortifications, or blocked by creatures, and all damage that would be dealt to it is prevented.

So nothing at all can target Progenitus.

"Protection from [a player] also is specifically defined:

702.16k “Protection from [a player]” is a variant of the protection ability. A permanent with protection from a specific player has protection from each object the player controls and protection from each object the player owns not controlled by another player, regardless of that object’s characteristic values. Such a permanent can’t be targeted by spells or abilities the player controls, enchanted by Auras the player controls, equipped by Equipment the player controls, fortified by Fortifications the player controls, or blocked by creatures the player controls, and all damage that would be dealt to it by sources controlled by the player or owned by the player but not controlled by another player is prevented.

Whether created by an emblem or not; the controller of the emblem is the controller of the triggered ability; so it cannot target True-Name Nemesis if that player was chosen.

One of the rulings below the card says

“Protection from everything” means the following: Progenitus can’t be blocked, Progenitus can’t be enchanted or equipped, Progenitus can’t be the target of spells or abilities, and all damage that would be dealt to Progenitus is prevented.

Teferi's emblem is a (triggered) ability, so I see no reason why Progenitus wouldn't have protection from it.

114.1. Some effects put emblems into the command zone. An emblem is a marker used to represent an object that has one or more abilities, but no other characteristics.

114.2. An effect that creates an emblem is written “[Player] gets an emblem with [ability].” This means that [player] puts an emblem with [ability] into the command zone. The emblem is both owned and controlled by that player.

603.3a A triggered ability is controlled by the player who controlled its source at the time it triggered, unless it’s a delayed triggered ability. To determine the controller of a delayed triggered ability, see rules 603.7d–f.

No

Quite simply, 'Protection from X' means (among other things) 'can't be targeted by X'. In the case of True-Name Nemesis, 'X' is the named opponent. The ability on the emblem says

"Whenever you draw a card, exile target permanent an opponent
controls."

The player can't target True-Name Nemesis because of its protection from that player.

Progenitus is an even simpler case. As others pointed out, Protection from Everything means (among other things), 'This can't be targeted'. The emblem's ability needs to select a target, so that cannot be progenitus.

If this were a case of the more common Protection from [a Color], the details about the emblem would matter. For instance even though Teferi is blue and white, as you mentioned, the emblem is colorless so the ability could target (for example) a Mistcutter Hydra