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After the 2018 PHB errata, the spell contagion now inflicts the Poisoned condition upon a successful melee spell attack. The relevant spell text is below, with updated portions highlighted in bold:

Your touch inflicts disease. Make a melee spell attack against a creature within your reach. On a hit, the target is poisoned.

At the end of each of the poisoned target's turns, the target must make a Constitution saving throw. If the target succeeds on three of these saves, it is no longer poisoned, and the spell ends. If the target fails three of these saves, the target is no longer poisoned, but choose one of the diseases below. The target is subjected to the chosen disease for the spell's duration.

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.

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 a target has poison immunity, they will still make Constitution saves as designated to see if the disease takes effect.

However, the line, "At the end of each of the poisoned target's turns..." also leads me to think that an enemy with poison-immunity cannot be the Poisoned target, therefore disallowing the remaining sequence of events.

Are targets with immunity to the Poisoned condition therefore immune to contagion?

Yes, they are unaffected

At the end of each of the poisoned target's turns

Since the target cannot be poisoned, there is no "poisoned target". The remaining description does not apply. If the spell is intended to still induce the disease even when the target has poison immunity, it would say "the target" or "if the attack hits, the target ..." instead of "the poisoned target". This is supported by Jeremy Crawford's reply to a Twitter user's tweet.

Twitter User: [...] ...On the other, it now makes it so poisoned immune creatures are immune the spell now as well.

@JeremyECrawford: Both things are intentional.

If a target is immune to disease, however, contagion does still apply poisoned condition to it until it fails or succeeds on all three saves. It does not get the effect of any of the diseases in the spell description if it fails three times, however.