I've found an article actually written by Robert Lacey.
I won't quote it all as it's really long but easily Googled:
It was not until this explosive material was in the can, being edited by a secret BBC team at a hotel in Eastbourne, that Diana started to reflect on how her two boys — both away at boarding school — might feel about it.
According to Simone Simmons, the Princess’s confidante and faith-healer, it took a phone call from William’s Eton housemaster, Dr Andrew Gailey, to prompt Diana.
Gailey had read the advance publicity in the newspapers and phoned to tell her it was ‘imperative’, in his view, that she should come to explain things to William, face to face.
‘Is that really necessary?’ she asked him.
In another phone call from Gailey the next day, Simmons told the editor-in-chief of royal magazine Majesty, Ingrid Seward, he effectively ordered Diana down the M4 motorway to talk to her son.
Afterwards she drove on to Ludgrove, Harry’s prep school half an hour away, to deliver the same message in person to her younger son.
‘Don’t worry,’ she assured him. ‘Everything will be fine — I promise.’
The 11-year-old Prince was offered the chance to watch the interview on the TV set of his personal protection officer at Ludgrove.
He declined, and the details of how Harry later reacted to his mother’s revelations are still obscure.
But William went down to his housemaster’s study shortly before 8pm that Monday to sit alone and watch as Panorama’s revolving globe and percussive theme music made way for the close-up of his mother’s wide eyes and heavily kohled eyelids as she fired off her broadside of embarrassing accusations.
Before the 58 minutes ended, William was weeping.
Gailey told Diana that he found her son slumped on the sofa, his eyes red with tears.
The Prince pulled himself together to rush back to his room — but when, an hour later, Diana telephoned on the house phone, William refused to take her call.
Something inside him had snapped. ‘He hated the idea of everything being on television,’ related Simmons, ‘and he knew his friends would poke fun at him, which they did.
‘He felt she made a fool of herself — and of him.’
**
By the time he went home to Kensington Palace at the end of that week to see Diana, he was raging. ‘All hell broke loose,’ Diana told Simmons the following Monday.
‘He was furious . . . that she had spoken badly of his father, furious that she had mentioned Hewitt . . . he started shouting and crying and when she tried to put her arms around him, he shoved her away.’
The asterisks show where I've cut out some bits about Eton traditions which didn't seem pertinent.
William's actions don't seem unreasonable for a 13 year old boy.
I haven't found anything on several ‘royal authors’ confirm[ing] William’s short temper/ bellowing rages.
(Letters in square brackets are mine. The apostrophe is not)