As you say: the game makes no such distinction between mundane and magical diseases with this ability, with all diseases just being diseases as far as it's concerned. Divine Health will protect you from both the measles and death dog saliva, equally.
This is congruent with how Paladins' miraculous healthiness works in many prior editions of the game too, so it's certainly not an oversight to not distinguish between mundane and magical disease.
Aside, the game otherwise does make a distinction between magical and mundane diseases (see the descriptions of spells like raise dead and resurrection) but Divine Health doesn't discriminate and defeats them all.
Not likely, but a case could be made for it
This is a bit of a tricky area, but in general we have go with the idea that things do what they say they do and no more. As with all things 5e, a DM can choose to view it differently, but looking at it through a RAW lens implies that the Paladin would still suffer the effects.
The case for Infestation
As I stated above, in 5e things (like spells) generally do what they say they do and no more. There is no clause in the Spawn of Kyuss that states that creatures immune to Disease are immune to the effects of the Spawn.
In addition, the Spawn of Kyuss doesn't state that it is a Disease. Just that using something that cures Disease will force them to wither away.
The case for Immunity
This piece comes from the viewpoint that much in the same way that a creature who is immune to certain conditions (like Charmed) makes them immune to spells that cause that such a condition.
There are a myriad of examples that show that being immune to a condition makes you immune to an effect (e.g., creatures immune to Frightened are not affected by a Vengeance Paladin's Abjure Enemy). Or that creatures immune to Charm are not affected by spells/abilities that Charm them.
The logic here is that if curing a disease kills the Spawn, and a paladin is immune to disease, then why would this 'disease' be active in it's body?
It's a reasonable step, but it isn't totally supported by RAW. A DM could houserule that as there is some sense to it, but I do not believe this to be RAW, however it may be intended.
Best Answer
Many types of internal parasite, including the egg of the red slaad and the worm of the spawn of kyuss, are explicitly described as being killed by any effect that removes disease.
However, pubic lice are homebrew and we don't know if they follow the same rules as the parasites listed above. You'll have to ask your DM.