Last member of the Royal Family to serve in action was HRH Prince Andrew, now the Duke of York in 1982.
Last member of the Royal Family to die on active service was HRH Prince George, Duke of Kent in 1942. He had relinquished the rank of Air Vice Marshall to become an Air Commodore so that he could actually do some real work.
The last King to have definitely served in combat was King George VI who as HRH the Prince Albert served aboard HMS Collingwood at the Battle of Jutland in 1915.
King Edward VIII (as Prince of Wales) was awarded the Military Cross in 1916, but it's unclear how much service he actually saw. It was probably some, though he wasn't supposed to be fighting.
Not to pick on your post, but the Royals really have served with honour in the military, and at risk to themselves at times. It's nothing special - thousands of others have done the same - but nor should it be viewed as unusual. They are raised to serve their country and many take that very seriously.
I had the honour to serve for a time alongside the Duke of York, and I can also say that the Prince of Wales did his duty during his much briefer service in the military from knowing men who served alongside him. The Duke of Edinburgh (before his marriage to the Princess Elizabeth, now the Queen) saw combat during the Second World War. Her Majesty served in the Auxiliary Territorial Service during that war.
Harry comes from a genuine military tradition, as well as a Royal tradition.