I haven't played around much with the culture, but EUIV's culture things work like this -
- You have a Culture Group, to which you belong
- All cultures in that group can be considered accepted, and any culture in any group cna be considered accepted if you are the primary nation of that culture group (France for the French culture group)
- Cultures outside of your culture group can be considered accepted based on the amount of provinces you have cored with that culture, and their base tax
So you can keep French culture, but this isn't the best option. For that matter, if you win the 100 Years War as England, I THINK you enter into a personal union with France. After 50 years, you can integrate France, which should give you all their culture's as accepted if you're lucky.
I don't think Irish can be an accepted Culture for England/Great Britain.