I don't know enough about the actual Prince John nor the history of the televised drama to know, in each scene, where historical fact leaves off and the drama begins. But my guess is that the Romanovs were incorporated not just because, yes, they did visit their Windsor cousins, but because dramatically they could be used to represent a unified but fragile family that would inspire both a sense of longing and, ultimately, sadness, for both Prince John and the viewer.
This much I know: The much-discussed episode of Alexandra and the shoes did not, in fact, occur--the man behind the drama admits this--but was used as a way of indicating that she could be difficult and cranky. If you look at Alexandra's face in that famous group photo, you will see that she was not happy to be there--not at all relaxed, as she was in so many photos taken during visits to her German relations--and I think that look of entrapment and discomfort was the scriptwriter's inspiration for the shoe episode. I also doubt, as I have expressed elsewhere, that Nicholas was quite the subserviant husband as portrayed in the drama; I find the portrayal of him and his attitudes towards Alexandra in the film of Nicholas and Alexandra much closer to what I've read in their correspondence, plus various memoirs and the like. In other words, when Nicholas found Alexandra to be a bit difficult, he generally ignored her behavior if they were in public--after all, she had ladies-in-waiting to assist her--then, if he did discuss the matter with her, did so privately. I think we need to remember that while Nicholas and Alexandra did have a great romantic relationship, theirs was also a relationship between two people who had lived together many years and were used to each others foibles and so forth. But they also were mindful of gossip and, despite Alexandra's obvious emotional issues, kept a great deal of their behaviors supressed while amongst others.
Getting back to the televised drama, I would agree that the scenes portraying the Romanov imprisonment were generally off-kilter. What I did like was Prince John's attraction to OTMA--that seems to me an absolute possibility, as practically all of Europe, not to mention the world, was entranced by these four beautifully dressed, seemingly ethereal young girls--and I also like Prince John's fantasy sequence, in which he imagines his Romanov cousins living the happy "farm" life as he lives.
The dramatist obviously was not in tune with those of us who have made a study of the Romanovs. What he was doing was using the Romanovs dramatically to put across some of his own points about Prince John. As a dramatist I think he succeeded; as an historian I think he did not.