How would the duke's children be referred to by the servants?

I think "young master" or "young lady/lord" would be quite accurate. Also naming the girl "Lady XXX" would also be fine, since royal/noble girls are often seen as ladies from the start who simply need training and education.
I heard that in many different time setting (like Victorian era in England or fantasy royalty) and different media (anime, movie and others).

I hope that was at least a little bit helpful. :3

loiterer said:

I googled, and came up with marquess for the boy, but it doesn't sound right to refer to a fifteen year old boy as a marquess--and I turned up nothing for the girl.

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I think Google is leading you astray. Courtesy titles and forms of address are completely different things. Assuming a 19c British baseline: "Marquess of XYZ" could be a courtesy title given to the boy IF he's the duke's oldest son, the duke possesses a marquisate in addition to his dukedom, and the title of marquess is customarily granted to the duke's oldest son. There's no age restriction on courtesy titles. But no one would go around calling him "Marquess" in conversation. (Though if he's marquess of Bigly his chums might call him Bigly.) As children of a duke the children would also be known as Lord X and Lady Y. The servants probably refer to them, as Sinuka says, as "young master" or "young lady/lord." But when speaking with them directly, it would be "my lord/lady" or "your lordship/ladyship".

If your setting is not 19c Britain or based on it, of course, the answer could be completely different. Everywhere makes its own rules for this sort of thing.

"Young master Edward" or "young mistress Charlotte" would convince me if used by the servants who had daily contact with them.
The heir to the dukedom could be a Marquess or an Earl depending on the tradition of that dukedom. The younger children are Lord or Lady.

Benbenberi is correct. In Britain the eldest son would be Lord [Courtesy Title]. The rest are Lord or Lady Firstname.

Whether the servants call them Young Master/Miss or Young Lord/Lady or Lord/Lady Firstname, or My Lord/Lady (to their faces) would depend on what the Duke and Duchess enforce or, if they have no strong feelings, then what the Steward, Butler or Housekeeper enforce. That would depend on how formal those people are--so ultimately up to you.

Ditto for the heir. Though if he's the Marquess of Queensbury or the Earl of Shrewsbury, for example, then at that age I'd find it a bit odd if they called him other than Lord Queensbury/Shrewsbury and "My Lord" to his face.

His family would probably call the heir Queensbury/Shrewsbury (no Lord) and only use his given name if no one else was around. The rest of the siblings would probably be called by their given names and referred to (to servants and people not close to the family) as Lord/Lady Firstname.
"Charlotte, darling, come sit here. Meriwether, bring Lady Charlotte her blue cashmere shawl."
"Queensbury, you're all in your dirt. Bunter, take his Lordship's coat."

This is assuming your story is not contemporary.

For the UK Debrett's Peerage is the resource.