Harry could only become Prince of Wales if his brother dies before their father. Even if William becomes King without yet having a son or daughter, Harry would only be Heir Presumptive - and only the Heir Apparent can be Prince of Wales.
If he does somehow become King - which is not absolutely impossible - he is at the moment, third in line - he would be Henry IX - the last Henry was still Henry VIII.
The Crown has been for the last several decades completely irrelevant to the health of the country and its people. If they do not assert their power very soon, it will result in the complete dissolution of their position and Buckingham Palace will be nothing but a crumbling relic.
The Crown is only supposed to assert its powers in particular situations of Constitutional crisis. None of these have arisen in recent years - if, for example, Gordon Brown, had decided not to resign after last years election, the Queen would have been compelled to act - but he did what he was supposed to do, so that was unnecessary. When the British system is functioning the way it is meant to, the Monarch is not meant to intervene.
There was a Henry IX who made a claim... from Scotland...
I would say the situation requires the monarchy to act, parliament is doing nothing about the Islamic invasion of Britain and the lawlessness.
Britain has become a Monty Python script... and it isn't funny anymore...
The movie “The King’s Speech” implied that a king can take a different name than a given name (or at least choose between given names). True?