The birthday of the Princess Royal 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/
North Riding Day takes place annually on 22 August.
It marks the date of the battle of the Standard (1138), when English forces defeated the Scots near Northallerton.
The name of the battle derives from the consecrated banners of the minsters of Durham, York, Beverley and Ripon.
These flew from a mast mounted on a cart that marked the centre of the English position.
East Riding Day takes place annually on 24 August.
It marks the birthday of Hull-born MP William Wilberforce (1759–1833), leader of the movement to abolish the slave trade.
Nottinghamshire Day takes place annually on 25 August.
By raising the royal standard at Nottingham Castle, King Charles I triggered the start of the Civil Wars.
Turks and Caicos celebrate Constitution Day annually on 30 August.
The date honours the first constitution of the islands, adopted on 30 August 1976.
Northumbria Day takes place annually on 31 August, the feast of St Aidan.
Aidan of Lindisfarne (d. 651) was a monk and missionary.
He founded Lindisfarne Priory and by tradition converted the Northumbrians to Christianity.
Merchant Navy Day – introduced by the UK government in 2000 – is not listed as a flag flying day by UK Government Guidance.
https://www.gov.uk/guidance/designated-days-for-union-flag-flying
Merchant Navy Day is listed as a flag flying day by Scottish government guidance.
The Red Ensign is specified.
https://www.gov.scot/publications/flag-flying-on-government-buildings/
Merchant Navy Day is also celebrated by the Senedd Cymru.
The Red Ensign is specified.
https://senedd.wales/media/hl5dtuct/foi-574-final-redacted-response-saes.pdf
The Accession Day of HM The King 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
The Accession Day of HM The King 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/
Gibraltar celebrates its National Day – first held in 1992 – annually on 10 September.
The date marks Gibraltar’s first sovereignty referendum (1967), when voters chose to stay under UK sovereignty rather than transfer to Spanish sovereignty.
Rutland Day takes place annually on 13 September.
It marks the defeat of the ‘Grand Assault’ during the siege of Gibraltar in 1782.
Among the defenders were the men of the 58th (Rutlandshire) Regiment of Foot.