Yes, Edward VI and other monarchs wrote diaries. Here are some extracts :
Edward VI, early 1547: “After the death of King Henry th’eight his son Edward prince of Wales was come to at Hartford by th’erle of Hartford and S[ir] Anthony Brown Master of t’horse for whom befor was made great preparation that he might be created Prince of Wales, and after ward was brought to Enfield whear the death of his Father was first shewed him, and the same day the death of his father was shewed in London, wher was great lamentation and weping and sodenly he proclaimed King. The next day, being the _ of _ he was brought to the Towre of London whear he taried the space of three wikes : and in the mean Season the Counsel sat every day for the performance of the performaunce of the will…”
22 January 1552: “the duke of Somerset had his head cut off upon Tower Hill between eight and nine o’clock in the morning”.
After Edward’s death, the habit took nearly three centuries to resume in British royal circles:
Victoria, 20 June 1837: “I was awoke at 6 o’clock by Mamma, who told me the Archbishop of Canterbury and Lord Conyngham were here and wished to see me. I got out of bed and went into my sitting-room (only in my dressing gown) and alone, and saw them. Lord Conyngham then acquainted me that my poor Uncle, the King, was no more, and had expired at 12 minutes past 2 this morning, and consequently that I am Queen”
11 February 1840: “I NEVER, NEVER spent such an evening!!! MY DEAREST DEAREST DEAR Albert … his excessive love & affection gave me feelings of heavenly love & happiness I never could have hoped to have felt before! He clasped me in his arms, & we kissed each other again & again! His beauty, his sweetness & gentleness – really how can I ever be thankful enough to have such a Husband! … to be called by names of tenderness, I have never yet heard used to me before – was bliss beyond belief! Oh! This was the happiest day of my life!”
July 1900: “Oh, God! My poor darling Affie gone too”, “It is a horrible year, nothing but sadness & horrors of one kind & another.”
George V and Queen Mary kept diaries too (visible here, particularly about Nicholas II‘s death), as did Elizabeth II, more privately.
I wonder whether Richard III ever wrote a diary?