The "Opportunity Attacks" section of the PHB/basic rules states (emphasis mine):

You also don‘t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

There is linguistic precedent in the rule that the word "move" does not expressly refer to the voluntary use of one's own Speed.

In the absence of specific wording to make sickening radiance an exception to the general ruling of nearly identically worded spells, the overwhelming likelihood is that it behaves like all the other spells described in the April 2016 Sage Advice rules answers column:

Does moonbeam deal damage when you cast it? What about when its effect moves onto a creature?

The answer to both questions is no. Here’s some elaboration on that answer.

Some spells and other game features create an area of effect that does something when a creature enters that area for the first time on a turn or when a creature starts its turn in that area. The turn you cast such a spell, you’re primarily setting up hurt for your foes on later turns. Moonbeam, for example, creates a beam of light that can damage a creature who enters the beam or who starts its turn in the beam.

[...]

Reading the description of any of those spells, you might wonder whether a creature is considered to be entering the spell’s area of effect if the area is created on the creature’s space. And if the area of effect can be moved—as the beam of moonbeam can—does moving it into a creature’s space count as the creature entering the area? Our design intent for such spells is this: a creature enters the area of effect when the creature passes into it. Creating the area of effect on the creature or moving it onto the creature doesn’t count. If the creature is still in the area at the start of its turn, it is subjected to the area’s effect.

Entering such an area of effect needn’t be voluntary, unless a spell says otherwise. You can, therefore, hurl a creature into the area with a spell like thunderwave. We consider that clever play, not an imbalance, so hurl away! Keep in mind, however, that a creature is subjected to such an area of effect only the first time it enters the area on a turn. You can’t move a creature in and out of it to damage it over and over again on the same turn.

In summary, a spell like moonbeam affects a creature when the creature passes into the spell’s area of effect and when the creature starts its turn there. You’re essentially creating a hazard on the battlefield.

The Fog Cloud cannot be colored, RAW

Without circumstances dictating otherwise, spells in 5th edition D&D only do what they say they do, and no more. The Fog Cloud spell says it creates a 20' radius sphere of Fog, and that's all you can guarantee it'll do. If you want it to do more than that, you'll need a different spell.

You should be able to combine Fog Cloud, Light, and (optionally) Mage Hand to create the appropriate effect

Placing a colored light source inside the Fog Cloud should diffuse the light through the fog, creating the colored Fog Cloud of your choosing. Since of those three spells, only Fog Cloud requires concentration, it's possible for the same spellcaster to maintain all three effects simultaneously, and both Light and Mage Hand are cantrips, meaning you're not spending extra resources (other than your Action Economy) to produce this effect.

Simply cast Light (for the color of your choosing) on a rock, then use Mage Hand to lift the rock into the fog cloud (or simply place the rock inside the fog cloud if your character is physically able to do so). The range on Light extends at least 20 feet, and while it wouldn't disable the Heavily Obscured effect, it is plausible that a character could generate a colored fog cloud using this method.

Bear in mind that you may need a permissive DM to make this work, RAW, since the fog will surround the light, and while this should not obscure the light source itself, it's not clear that the physics of the light diffusing through the fog will result in the fog itself taking on the light's color. So it does require your DM to allow for real world physics to take precedence in this situation.