King Charles speaks with a rather posh Received Pronunciation, much like Queen Elizabeth II did. In the meantime, William and Harry use a more Standard Southern British pronunciation.

The changes described by Lindsey can be summed up as:

• PRICE - [aɪ] vs. [ɑɪ]
• DRESS - [e] vs. [ɛ]
• CHOICE - [ɔɪ] vs. [oɪ]
• SQUARE - [ɛə] vs. [ɛ:]
• HAPPY - [ɪ] vs. [i]
• [ɫ] vocalisation, colouring nearby vowels - negligible vs. noticeable
• word ending /t/ - [t] vs. [ʔ]
• /t/ flapping into [ɾ] - rare vs. more frequent
• /t/ before front high vowel affricating into [ts] - actually attested for both sides
• unstressed syllable elision - King Charles did this quite a bit before rising to the throne, but William does it all the time
• rising intonation on statements (uptalk) - almost non-existent in RP, fairly common in SSB
• /θ/ as [f] - avoided in RP, present in SSB
• word ending /k/ as [k’] - avoided [?] vs. common

Personal observation: the changes in the vowel sets remind me in spirit the Great Vowel Shift, as it seems that DRESS lowering is pressing PRICE to go back, and in turn PRICE is forcing CHOICE to raise.