Say a second level paladin gets a disease, then at the end of the session levels up and gets Divine Health, which makes them immune to disease. Do they still carry that disease and are its markings visible?
The rules as written don't say specifically.
"Immune" doesn't carry any special meaning in 5e. An ordinary language interpretation of "immune" would be that a character immune to disease can't contract any diseases, and that any diseases they have are immediately cured.
The DM may rule otherwise, although at the very least it would seem reasonable that such a character would not suffer any further negative effects from the disease, and that they would automatically succeed at any future saving throws the disease calls for.
It is also up to the DM to rule if immunity to diseases removes any scars or blemishes that previous diseases may have inflicted, or if more serious permanent effects of diseases (for example, blindness from sight rot) are healed.
My reading of the rules as intended is that 3rd level paladins gain an advantage in the 'exploration' pillar of the game by ignoring a specific class (diseases) of environmental hazards. As such, I would expect that once they gain 3rd level, any negative effects of diseases are completely removed.