Trying to create an overview - please feel free to share corrections or additions.
The Netherlands:
Willem-Alexander: Dutch, English, German, Spanish, French
Máxima: Spanish, English, Dutch, French, (some) German
Amalia: Dutch, Spanish, English, German, (some) Mandarin (and probably also some French)
Belgium:
Philippe: French, Dutch, English, Italian, Spanish, (some) German,
Mathilde: French, English, Dutch, Spanish, Italian
Elisabeth: French, Dutch, English
Luxembourg:
Henri: French, Luxembourgish, German, English
Maria Teresa: Spanish, English, French, Luxembourgish
Guillaume: Luxembourgish, French, German, English, Spanish
Stephanie: French, English, Russian, Luxembourgish, German, (some) Dutch
Denmark:
Frederik: Danish, French, English, German
Mary: English, Danish, (some) French
Christian: Danish, English
Norway:
Harald: Norwegian, English
Sonja: Norwegian, English, French
Haakon: Norwegian, English, German
Mette-Marit: Norwegian, English
Sweden:
Carl-Gustaf: Swedish, English
Silvia: German, Portuguese, English, Spanish, Swedish, French
Victoria: Swedish, English, German, French
Daniel: Swedish, English, French, German (at least he got tutoring sessions in those languages)
Spain:
Felipe: Spanish, English, Catalan, French
Letizia: Spanish, English
Leonor: Spanish, English, (some) Mandarin
United Kingdom:
Charles: English, French, German, Welsh (basic)
Camilla: English, French
William: English, French (basic), Welsh (basic)
Catherine: English
Monaco:
Albert: French, English, German
Charlene: Afrikaans, English, (some) French
Jacques: French, English
Liechtenstein:
Hans-Adam: German, English, French
Alois: German, English, French (?)
Sophie: German, English, French (?)