The birthday of the Duchess of Edinburgh (and Countess of Forfar) is an official flag flying day only in Scotland, not in the rest of the UK.
https://www.gov.scot/publications/flag-flying-on-government-buildings/
Commonwealth Day is the second Monday in March.
The UK government states that the Union Flag should be flown.
https://www.gov.uk/guidance/designated-days-for-union-flag-flying
The Scottish government states that the Commonwealth flag should be flown.
https://www.gov.scot/publications/flag-flying-on-government-buildings/
The Senedd Cymru states that the Commonwealth flag should be flown.
https://senedd.wales/media/hl5dtuct/foi-574-final-redacted-response-saes.pdf