Alix was a vegetarian, while Nicholas had a fondness for Russian food: salted cucumbers, kvass, kashi, borshcht, shchi, and caviar (though it didn't agree with him) to name a few. There's also an anecdote about Alix crunching English biscuits (cookies) in bed, much to the emperor's annoyance, but I think it comes from from Marfa Mouchanow's book, which is NOT realiable at all.
As I recall, the food in exile wasn't anything special: meat cutlets, soup, macaroni, etc. The Grand Duchesses eventually learned how to make bread. A nearby convent brought eggs, milk, and maybe yogurt as well.