Of course, the flag of St. Patrick forms part of the Union Flag of the United Kingdom.
From when it was introduced by Richard I in 1149 AD, the flag of England the flag of Saint George. This is a red cross on a white background (familiar to us all from its use by England football fans).
In 1536, Henry VIII passed an act of union making Wales a province of England. As Wales was a principality and not a country, no changes were necessary to the flag of England.
After Queen Elizabeth I died, King James VI of Scotland inherited the English throne and became King James I of England. James I referred to his two kingdoms as Great Britain. A problem arose of which flag to fly on the king’s ships. In 1606, the problem was solved with a compromise resulting in the creation of the first Union flag.
The flag of Scotland was a white diagonal cross on a blue background (Saint Andrew’s cross). This was combined with the English flag to make a combined flag. A white border was added around the red cross for reasons of heraldry, the rules demanding that no two colours must ever touch. This was known as the British Flag or the flag of Britain.
After the Act of Union in 1707, England and Scotland were joined together with a single Parliament and called the United Kingdom of Britain. At this time the flag became known as the Union, usually the Union Jack as it was still mostly used on navy ships.
The Act of Union of Ireland with England and Scotland in 1801 required the addition of the cross of Saint Patrick (a red diagonal cross on a white background) to the Union Flag of Saint George and Saint Andrew to create the Union Flag that we still use today.
As you have pointed out, the cross of Saint Patrick is not centred. This is because it would have laid exactly on top of the cross of Saint Andrew implying that Ireland was superior to Scotland. However, since Scotland joined the Union before Ireland did, it was considered senior. This is signified by the St Andrew cross appearing above the cross of St Patrick "at the hoist" (the staff), the higher ranking part of the flag.
See these links for more information:
I guess there would be no need to remove the Irish flag to include the flag of St. David. It depends of course on what decision would have been made in 1536, but the flags of St.George and St. David are the same but different colours. They almost certainly would have put the red English cross on top with the St. David's cross showing around the sides. If they kept the white background, they could have then added the other two flags as they became included. This would look somewhat like this picture.
The red diagonal might have gone all the way to the middle, it depends on the exact rules applied. As you can see from the fuss about the Irish flag, these can be fraught with problems.
If the flag of David replaced the Irish flag, it would be like this but without the diagonal red bands.