It's one damage roll, it's applied to all characters.

How I do this in my games is have the damage rolled before anyone makes their save. So the caster determines the damage before the saves are rolled.

Then each character makes their save and the effects are adjudicated after the saving throws are made.

Page 196 of the PHB has this rule:

If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them....

The warship takes all 50 damage

In the quote you provided, it says (DMG, pg. 246):

An object with a damage threshold has immunity to all damage unless it takes an amount of damage from a single attack or effect equal to or greater than its damage threshold, in which case it takes damage as normal.

"It takes damage as normal" means that it would take the 50 damage as though it didn't have the threshold. The only difference here between objects with and without thresholds is that attacks that do less than the threshold do nothing.

This is not the same as, say, temporary hit points, which use a different wording (PHB, pg. 198):

When you have temporary hit points and take damage, the temporary hit points are lost first, and any leftover damage carries over to your normal hit points. For example, if you have 5 temporary hit points and take 7 damage, you lose the temporary hit points and then take 2 damage.

This talks about "carrying over" hit points. If that is how damage thresholds were supposed to work, similar wording would likely have been used. As it stands, it simply says that it "takes damage as normal", which means, from your example, the warship takes the full 50 points of damage.