The feature tells you if it is a disease.
This search on D&D Beyond shows numerous monsters with disease features. For example, the diseased giant rat’s bite says:
Bite. Melee Weapon Attack: +4 to hit, reach 5 ft., one target. Hit: 4 (1d4 + 2) piercing damage. If the target is a creature, it must succeed on a DC 10 Constitution saving throw or contract a disease. Until the disease is cured, the target can’t regain hit points except by magical means, and the target’s hit point maximum decreases by 3 (1d6) every 24 hours. If the target’s hit point maximum drops to 0 as a result of this disease, the target dies.
This is just one of many, many monsters with disease features. The linked search overloads DDB’s search capacity at 35 results.
Similarly, the spells harm and contagion tell you they inflict diseases:
You unleash a virulent disease on a creature that you can see within range. […] Any effect that removes a disease allows a creature's hit point maximum to return to normal before that time passes.
Your touch inflicts disease.
[…]
Since this spell induces a natural disease in its target, any effect that removes a disease or otherwise ameliorates a disease’s effects apply to it.
We also see diseases in published adventures clearly stated to be diseases. For example, we see in Tomb of Annihilation a section on the diseases you can contract in the jungles of Chult:
The following diseases can affect giants and humanoids exploring the jungles of Chult. Remember that lesser restoration and similar magic can cure a disease.
It then describes the effects of several diseases. Similarly, the adventure Frozen Sick published in Explorer’s Guide to Wildemount is about a special disease, frigid woe:
Frigid Woe. Frigid woe is a special disease developed by Aeor's mages that cannot be cured by conventional treatment or magic. The only way a creature infected with the disease can be cured is by finding and drinking the manufactured antidote, a milky liquid stored in gold vials found in Eiselcross's ruins. This disease was created to slow down the forces of the gods and get around the healing power of their clerics and angels.
There’s no rule or definition for disease by which we might discern if a particular effect is a disease, so it is quite natural to assume that a feature is a disease only if it states it is a disease, as all of these examples have done.
Disease curing magic works on it: maybe it’s a disease.
Things get a little bit more grey when a feature can be cured by disease curing magic, but it isn’t explicitly called a disease. For example, rot grubs:
Bites. Melee Weapon Attack: +0 to hit, reach 0 ft., one creature in the swarm’s space. Hit: The target is infested by 1d4 rot grubs. At the start of each of the target’s turns, the target takes 1d6 piercing damage per rot grub infesting it. Applying fire to the bite wound before the end of the target’s next turn deals 1 fire damage to the target and kills these rot grubs. After this time, these rot grubs are too far under the skin to be burned.
If a target infested by rot grubs ends its turn with 0 hit points, it dies as the rot grubs burrow into its heart and kill it. Any effect that cures disease kills all rot grubs infesting the target.
Does this count as a disease? Maybe. Personally, I would rule that immunity to disease would not protect from this, because it doesn’t say it’s a disease, and your über-robust immune system isn’t going to do anything to protect you from larger parasites like flesh burrowing grubs. But it is perfectly reasonable for the DM to rule otherwise.
Disease curing magic generally should not break curses.
Lesser restoration, which cures diseases, is a 2nd level spell. However, remove curse is a 3rd level spell. Ergo, curse breaking magic is generally different and stronger than disease curing magic. The mummy rot curse is a curse, not a disease, and lesser restoration is not a “similar magic” to the remove curse spell - it’s weaker. So the paladin’s lay on hands feature should not cure mummy rot.
Best Answer
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.