Well if we're being pedantic, definitely not English - there is no such "nationality". Czech-British? Maybe just "British" being the term used for citizens of the United Kingdom of Great Britain and Norther Island?
Moreover, England, Scotland, Wales, and Northern Ireland are recognized as constituent countries of the United Kingdom. (And is should be noted that “Great Britain” only encompasses England, Scotland, and Wales—not N. Ireland. So it’s definitely incorrect to refer to a N. Irelander as “British.”) They aren’t simply states in the way Texas or Tennessee are. They are countries with devolved governments, and are recognized as such by the UK government. As such, it seems entirely fitting, even from a political and pedantic standpoint, to refer to someone as belonging to the particular country from which he hails.
E: Probably also worth noting, since we’re being pedantic, that OC’s comment that “the man is quite English” need not refer to his nationality per se. It could just as easily mean that Stoppard is culturally “quite English” rather than indicating which national gov’t issues his passport.
That's a complex issue. Many Northern Irish people do consider themselves to be British and others consider themselves to be Irish. As you might expect, their preferred identity is often related to whether they are members of the Protestant/Unionist or Catholic/Nationalist communities (this is a significant oversimplification).