The question is almost meaningless.
As a British National, one has the right to live in Britain.
Does the Queen have the right to live in Britain? She only enjoys this privilege as long as the Government choose to allow it. There are lots of examples of Kings and Queens being deposed and subsequently exiled. It seems reasonably clear that while normal Britons enjoy the protection of the Queen's law, the Queen doesn't have these benefits as a right, but rather can remain in Britain while there is nobody strong enough to depose her.
In fact the whole notion of "rights before the law" breaks down when you have an individual person who embodies the power of the state.
This makes questions of the form "is the Queen a Canadian National" fundamentally meaningless. No situation could arise in which the queen could seek to exercise her potential Canadian Nationality (as distinct from her Canadian suzerainty) in which there had not been a major constitutional upheaval and her status as Queen had fundamentally altered.