1936 saw the death of George V, and the monarchy passed to Edward VIII who did not attend a coronation as he abdicated for the love of Mrs. Simpson, passing the throne to Bertie, Geroge VI, father of our present Queen.
Strictly there is no such thing as "the British throne" the Queen is Queen of Great Britain and Northern Ireland ; ie the United Kindom.
To refer to the current monarchy as solely British is technically incorrect.
and to carry on nitpicking......... The present monarch is Queen Elizabeth ll of England Wales and Northern Ireland only. In Scotland she can only be Elizabeth l
Of course the queen has the right to call herself whatever she chooses including the number.
The "II" suffix is used in all countries on the commonwealth regardless of any link or not with the first Queen Elizabeth, because that is her chosen title.