Why did Nicholas and Alexandra write to each other in English?
Alexandra (born Alix of Hesse) was originally from Germany, born to a German father, and an English mother. It's important to note her mother was Princess Alice, daughter of Queen Victoria. Alexandra spent time visiting her family in England throughout her early life and was given an English and German education.
After marrying Nicholas, her Russian made her unpopular with the locals. Unfortunately, I cannot attest to her fluency, so I am not sure whether she couldn't speak and write it well or if she was uncomfortable speaking it, or a combination of both. Regardless, there was a degree of unease with the empress and the Russian language.
Nicholas was tutored in many languages including Russian, English, German and French. His English tutor, Charles Heath, held great influence over young Nicholas resulting in his English being very good. In correspondence between Nicholas and Kaiser Wilhelm II of Germany, they both used English. I'll link to these messages below (called the "Willy-Nicky Correspondence").
In truth, it can't definitively be said why English was the chosen language for Nicholas and Alexandra. However, we do know that they were both fluent in English and did not have the same fluency and/or comfort level in their other "shared" languages.
Was it uncommon for continental royalty to know English?
For centuries, it’s been usual for royals to know more than one language. Even today, Queen Elizabeth II can speak English and French. Further back, we see individuals like Joanna of Castile, Elizabeth I, and Louis XVI of France who were all taught a multitude of languages. Some general reasons for this can be found in /u/Trinity-‘s comments in this thread and /u/AndrewSshi’s comment. The main idea is that knowing many languages was useful for royals.
I don't think I can give you a thorough answer. I wouldn't know how to define "uncommon" to begin with. What I can tell you is that the British Empire in the 19th and early 20th centuries was at its height. We have examples of continental royalty descended from Queen Victoria who spoke English (Empress Alexandra, Kaiser Wilhelm II, Sophia of Prussia to name a few) and we know Tsar Nicholas II spoke English fluently despite not having immediate English ancestry. Christian IX of Denmark's children were given an English education as well (one would eventually marry the King of England) without having immediate English ancestry either.
Sources:
Scenarios of Power: Myth and Ceremony in Russian Monarchy by Richard Wortman
Edit: rewording, grammar and clarifying