The Edinburgh Tattoo started out as a military contribution to the Edinburgh Festival and followed the format of similar events such as the Royal Tournament. It has evloved over the years to what it is today.
While International in flavour it is still very much a British military event with as much emphasis as possible on the fine Scottish military traditions. National Anthem is therefore very appropriate as are also military hymns such as Evening Hymn (sunset ceremony).
The BBC broadcast is normally filmed over several nights and then editted together to make a pleasing programme. I very much doubt there was any sinister underlying Scots Nationalist agenda in the editting.
Overall it is a great event, raises considerable sums for Service charities and others and plays a strong part in keeping the Services in the public eye. It should be receiving our praise not some little Englander angst about whether the Anthem was broadcast on TV.