Diplomats are senior Foreign & Commonwealth Office employees. For careers information, start here:
http://www.fco.gov.uk/en/about-the-fco/working -for-us/
MEPs are representatives of political parties, in the same way that most MPs are. They normally start out by taking an active part in local party politics (e.g. by helping with election campaigns) and then stand, on behalf of their party, for election to local councils. The high-flyers often go on to serve on their county council before being nominated for a parliamentary seat. Some politicians receive their party's nomination directly to the European Parliament but quite a few MEPs have served as MPs, in the House of commons, before they're selected to stand for the European Parliament. MPs and MEPs are drawn from a wide range of backgrounds (and thus have a wide range of degrees) but, for a career in politics, a degree in political science or economics might be provide a slight advantage.
Additionally, the European Parliament and the European Commission both employ their own staff. The senior staff will normally be selected because of their specialist knowledge of banking, finance, economics, international law, etc but it might also help their careers if they're at least bilingual (English and French) or preferably fluent in several languages.
Chris