Can the British Royal Family be arrested?

Technically, yes, apart from the Queen. The courts are her courts, the police force serve her, prisoners are detained under her pleasure. If another member of the royal family was arrested, she can't demand a release, but she herself is technically immune from prosecution because she 'is' the law.

Now if the reigning British monarch did commit a major crime, something overt and obvious such as a murder, then parliament will have to decide how to handle it. The main thing I'd say is, because it has never happened and is rather unlikely to happen, no one has seen the need to put a procedure into law.