St David’s Day is Wales’ national day.
The Senedd Cymru flies the Welsh national flag – Y Ddraig Goch (The Red Dragon) – all year round.
On St David’s Day it also flies the flag of St David (below).
https://senedd.wales/media/hl5dtuct/foi-574-final-redacted-response-saes.pdf
St David’s Day is also listed as a flag flying day (for Wales only) by UK government guidance.
The Union Flag is specified.
https://www.gov.uk/guidance/designated-days-for-union-flag-flying
St Patrick’s Day is listed as a flag flying day (for Northern Ireland only) by UK government guidance.
The Union Flag is specified.
https://www.gov.uk/guidance/designated-days-for-union-flag-flying
There is no official, current pan-Ireland or Northern Ireland flag that may be used to mark St Patrick’s Day.
St Patrick’s Saltire is one of several possible alternatives.