All of us who work in finance and private wealth should be concerned about corruption and money-laundering still taking place in the industry. Most of us believe that promoting artificial avoidance schemes doesn't do anyone any favours and some of us may even think the wealthy can afford to pay a little more tax! The problem with the BBC story is that it encourages moral panic based on superficial and ill-informed reporting. If cross border finance in itself is the evil then this doesn't bode well for prosperity! The story about the Duchy of Lancaster in itself is ridiculous because the Duchy is under Government control, the investments were ultimately in UK businesses, they were structured through jurisdictions (the Cayman Islands and Bermuda) under indirect UK government control and with the Queen as head of state, and the Duchy and Queen have a unique tax status. As for Appleby, the leaks seem to have shown the firm in rather a good light, with a concern that structures were managed properly and due diligence processes adhered to.
There is nothing illegal in the investments and no suggestion that the Queen is avoiding tax, but questions may be asked about whether the monarch should be investing offshore.