After the PHB 2018 errata, the spell contagion now inflicts the Poisoned condition on a successful melee spell attack. The text of the relevant spell is below, with the updated parts highlighted in bold:
Your touch inflicts diseases. Make a melee spell attack against a creature within reach. In one hit, the target is poisoned.
At the end of each turn of the poisoned target, the target must make a Constitution save roll. If the target achieves three of these saves, he is no longer poisoned and the spell ends. If the target fails three of these saves, the target is no longer poisoned, but chooses one of the following diseases. The target is subject to the chosen disease for the duration of the spell.
Since this spell induces a natural disease in its target, any effect that eliminates a disease or that otherwise improves the effects of a disease applies to it.
The wording of the spell leads me to conclude that this spell still inflicts a disease, which has the immediate effect of poisoning the target. Therefore, even if an objective has immunity to the poison, they will continue to make the liftings of the Constitution as designated to see if the disease has an effect.
However, the line, "At the end of each turn of the poisoned target …" It also leads me to think that an enemy with poisonous immunity can not be the target poisoned, therefore, does not allow the remaining sequence of events.
They are targets with immunity to Poisoned condition therefore immune to contagion?