I compared the wording of the spell to some other spells that have saving throws and multiple effects. If the extra effect only takes hold on a failed save, that is often stated. Some examples:
Reality Break (EGW, p. 189):
The target must succeed on a Wisdom saving throw, or it can't take reactions until the spell ends. The affected target must also roll a d10 at the start of each of its turns; the number rolled determines what happens to the target, as shown on the Reality Break Effects table.
Whirlwind (XGE, p. 171)
A creature must make a Dexterity saving throw the first time on a turn that it enters the whirlwind or that the whirlwind enters its space, including when the whirlwind first appears. A creature takes 10d6 bludgeoning damage on a failed save, or half as much damage on a successful one. In addition, a Large or smaller creature that fails the save must succeed on a Strength saving throw or become restrained in the whirlwind until the spell ends.
So one could argue, if the exhaustion would be conditional on failing the saving throw, that would have been stated.
I had a hard time finding a spell where an extra condition is explicitly not dependent on the saving throw, however I found one:
Temple of the Gods (XGE, p. 167):
If a creature of the chosen type attempts to enter the temple, that creature must make a Charisma saving throw. On a failed save, it can't enter the temple for 24 hours. Even if the creature can enter the temple, the magic there hinders it; whenever it makes an attack roll, an ability check, or a saving throw inside the temple, it must roll a d4 and subtract the number rolled from the d20 roll.
So here one could argue if the exhaustion would be independent from the failed save, that would be stated explicitly.
Both Whirlwind and Temple of the Gods are from the same source book as the Sickening Radiance, so it seems reasonable to expect the wording to be similar.
Therefore I conclude: The wording is unclear.
It's not a very useful conclusion, but as I already did the research about other spells wordings, I at least wanted to note it down, if only as a base for other people's answers.
Personally, I think the exhaustion and glow is only on a failed saving throw. The spell is already reasonably strong like this; if you make the exhaustion not dependent on the saving throw it, it's overpowered.