The Birthday of the Prince of Wales is listed as a flag flying day by UK government guidance.
The Union Flag is specified.
https://www.gov.uk/guidance/designated-days-for-union-flag-flying
Under his Scottish title (Duke of Rothesay), the Birthday is also listed as a flag flying day by Scottish government guidance.
The Scottish Saltire is specified.
https://www.gov.scot/publications/flag-flying-on-government-buildings/