Effects with different names can affect a creature at the same time.
In the Dungeon Master's Guide on page 252, the rules for combining game effects say:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the “Combining Magical Effects” section of chapter 10 in the Player’s Handbook.
So these two features, Necrotic Shroud and Wrathful Smite, can affect a creature at the same time. What does this mean? It does not mean that the creature is twice as frightened, the frightened condition is only meaningfully applied once, but it means that both sources of the effect must be resolved before the creature is no longer frightened of the source of fear.
Guidance on simultaneous effects from Xanathar's Guide to Everything on page 77 suggests that the player (or the DM) controlling the creature gets to choose which save to roll first:
Most effects in the game happen in succession, following an order set by the rules or the DM. In rare cases, effects can happen at the same time, especially at the start or end of a creature’s turn. If two or more things happen at the same time on a character or monster’s turn, the person at the game table — whether player or DM — who controls that creature decides the order in which those things happen. For example, if two effects occur at the end of a player character’s turn, the player decides which of the two effects happens first.