I'm not sure about which language was used all the time, but since you mentioned George, Wilhelm and Nickolas, I believe this is relevant:

Before the outbreak of WWI, Wilhelm of Germany and Nikolas of Russia sent eachother a series of very polite telegrams basically trying to talk eachother out of this whole war business. These were known as the Willy-Nicky correspondence, because that's how the emperors referred to themselves in them. According to this, these conversations were in English, which was the two's "common language".

I recently read 'from cradle to crown' by Charlotte Zeepvat which is all about British (and Irish) nannies and governesses who worked in Royal Courts throughout Europe, Russia, Asia, the Middle-East, the Far-East and even South America from the 18th c. to the 20th c.

Which I recommend for anyone who's interested in the inner workings of Royal Courts, Royal childhoods or the changing role of women and how typically female roles such as child-care became professionalised.

Anyway, I was surprised by the extent to which English was the dominant second, often even first, language of many of these Royal families prior to Queen Victoria's reign. As well as during and after it.

The Russian Royal family employed British (or Irish) nannies in the late 18th c. and that set a 'keeping up with the Jones' ' fashion for having Royal children educated in English by a native speaker, so they're often learning English as their 'mother' tongue (because they are essentially being raised by nanny) but not their national language until they start formal schooling (or the private equivalent, since they are usually educated at home). These nannies were usually part of a network so the same women would move around different Courts as children grew up and new ones appeared.

As these careers became professionalised in Britain that created a feedback loop which increased demand from foreign Courts.

There is a stereotype which emerges, I know not whence, but it is sometimes mentioned in private correspondence when Royal parents and staff are choosing nannies, that British, particularly English, women have a greater maternal instinct and therefore make better teachers and child-minders. I suspect this idea comes from people who were themselves raised by British nannies.

The process of English being the Lingua Franca of Royalty is accelerated by the growth of the British Empire and by Queen Victoria's fecundity later, but it doesn't start there. It starts in the nursery much earlier.

By the late 19th. c. the British (or Irish) nanny is an expected fixture in Royal Courts, especially in Europe, but further afield as well.

Wilhelm II and Nicolas II both had British nannies and both could write English fluently, so there would not have been any translation issues. I'm not sure if George V could speak German or Russian, but it's mostly likely they would have used English amongst themselves.

Just to be clear, English isn't the language of court, that's usually French or the national language. English is the language that's being used when they are interacting with each other as private family members, or with their closest members of staff, as opposed to their public or Royal roles.

Wilhelm and Nicolas sent each other a number of letters and telegrams, the most notable (and in some ways quite chilling) being the telegrams sent during the political maelstrom before the Great War. You can find a scan of an original text here and read the pertinent telegrams here.

These letters seem to have been sent exclusively in English, indeed, Nicolas spoke English at home with his German wife. Whilst I'm not sure of any specific evidence, it's a reasonable assumption that their preferred informal, shared language was English. They certainly wouldn't have needed interpreters.