For those wondering why everything is being removed...

I believe . . .

Not an expert, but . . .

That is sheer speculation, and thusly doesn't count as a scholarly answer.

They spoke French or Latin . . .

They were rich, and that means they were educated and learned several languages . . .

This is not in-depth and comprehensive.

What's with all the fucking removals?

This is neither polite nor on topic.

It depended on the marriage, and it depended on the people involved.

In Henry and Catherine's case (or rather, Arthur and Catherine's case since Catherine could speak English by the time she married Henry), they both spoke French. However, they had learned it with different accents so communication was still difficult. They had written letters to each other previously in French.

Catherine of Aragon was extremely well educated, with her education being personally overseen by her mother, Isabella of Castile. By the time she came to England, she was fluent in Spanish, Latin, French, and Greek. Arthur would have been fluent in English, Latin, and French at the very least. Catherine eventually learned English, but spoke with a heavy Spanish accent for the rest of her life.

Anne of Cleves, on the other hand, had a very poor education (in comparison to Henry's previous marriages. In comparison to the average woman of the day, it was rather good seeing as how she could read and write) and could only speak German. In contrast to Catherine of Aragon, Anne was not raised with the assumption that she would someday be queen of England, and so no one bothered to teach her French or English, and because she was a Lutheran, Latin was out.

When she came to England, she had several interpreters with her. She did eventually learn English, but until that time, she and Henry spoke to each other through interpreters.

In a normal marriage situation, negotiations would take place while the prince/princess were children. Often, the girl would be sent to be raised in the court of her betrothed until they were old enough to be married. She would learn the language there. This is what happened with Mary, Queen of Scots, although to be fair, had she stayed in Scotland her normal education would have included learning French since Scotland and France were ancient allies (the alliance being nicknamed "The Auld Alliance").

Even if this didn't happen, many of the European courts spoke French. When Juana of Castile married Philip the Handsome, Philip was from the court of Bergundy, so while he was the son of the Holy Roman Emperor, it was a French court. Juana was sister to Catherine of Aragon and had the same education, meaning she could speak French as well.

When Henry VI married Margaret of Anjou, she was again a French princess and could easily adapt to the French speaking English court.

Anne Boleyn would have learned French as part of her education and practiced it when she served Queen Claude of France as a maid of honor.

Primary source: The Six Wives of Henry VIII by Alison Weir

Related but aside: Is there a reliable but layperson-accessible biography of Catherine of Aragon?