I like the Dutch derivative (John - pronounced Yan, Cornelius - pronounced Kee) but I have no idea if it's correct.
Anyway, some people here are aware that it can have derogatory overtones and will avoid using it. I can happily say I am one of those.
Be wary though, as throughout the South East you may hear the word, 'Septic' when referring to US citizens (Cockney Rhyming Slang: Septic Tank / Yank, reduced to 'Septic'). Definite derogatory overtones there.
Similarly, you may sometimes (almost exclusively in London) hear people refer to themselves or others as 'Listerine'. Listerine is a mouthwash, known for it's antiseptic action and (distressingly) it's aftertaste. Anti-septic = one who dislikes Americans. Usually without knowing any, but hey ho.
If there's one area where Britain still leads the world it's in casual (and often amusing and/or deprecating) insults aimed at ourselves or others. Personally, I think we have the French to thank for that.
So people would know who they were talking about. You can't run a world calling everybody "these guys over here" and "those guys over there". They gotta have names.
I may be wrong but I think the appalling contraction 'Brits' was invented and certainly encouraged by British journalists writing for The Sun or similar low-quality newspapers.
I'll let BHWilson explain 'Limey' but will say that although it has been used sarcastically by anti-British Americans, it does have an innocent, indeed, honourable origin.
"Britons" is an invention based on "Britain", which was an invention dated 1707, based on "British Isles". Nobody knows why they were called "British Isles", but that is what they have always been called.
The name "England" came about when settlers from Belgium came in. They were fishermen, and their word for a fish hook was "Angul", and they were called "Angulers", and they called their new home "Angul Land". That also explains why fishing is called "angling". There is still a province in Belgium called Angul.
I can't find a youtube clip, but the story goes like this. The Navy knew lemons were more effective for scurvy, but due to the (then) current political situation with France/Spain/Holland the British government had no easy access to lemon growing areas. So, they opted for limes as the next best thing. So, but for politics the nickname might be 'lemoneys'.
I'm not convinced by Jules Vern's un-cited source for the word "Angles". It has nothing to do with angling, despite that perhaps having a historically relatively recent Belgian root. For centuries prior to angling with hooks, fishing was by net and for food, not sport.
"England" was initiated by the Anglo-Saxons, hence the name. Wales, Scotland and Ireland were still their own kingdoms.
The Angles, Saxons, Jutes and others of the time were invaders and immigrants from what is now mainly Germany and the Scandinavian / Baltic countries, after the end of the Roman era.
The Anglo-Saxons started to run the country in a formal manner but as a set of principalities or minor kingdoms with a fairly uniform religion. They laid down early laws, and started building stone churches of which traces linger in some despite later rebuilding over many centuries.
It was not a very peaceful time: the Vikings invaded in turn at various times, leading sometimes to fierce fighting between the established Anglo-Saxons and these Scandinavians. ("Vik" is still Norwegian for "bay" and occurs in many Norwegian coastal place-names.) Most of the Vikings landed in the North-East of England, and their language's influence survives strongly in what is now Yorkshire.
One resulting mass grave found during road-building excavations several years ago in the Southern English coastal county of Dorset, held what are thought to have been Vikings, massacred by the resident Angles barely 4 miles inland from the sea forming part of the view from the hill there. The dead had all been beheaded and their heads buried together to one side of the bodies, but whether by execution or a post-mortem insult by victor of vanquished dead, we cannot know.
Their works introduced the Mediaeval Period, well before the Norman invasion in the 11C; the Norman invaders crossed from France but were Norse, i.e. of Nordic (Scandinavian) descent.
"Britain" as a name could only arose when all four kingdoms became the United Kingdom. In Ireland's case, under William of Orange after he'd seized power in England. That coup and its inbuilt Protestant/Catholic schism, rankles today with many Irish, background to the independence of largely-Catholic Eire in 1916, and the euphemistically-called "Troubles" in Ulster in the 1970s. England had become officially Protestant under Henry VIII, and after a period of mutual fear, hatred and persecution, learnt to accept Catholicism as just another form of Christianity. However, the Vatican had lost the political powers it had held in what became Britain since Anglo-Saxon times. Although the UK's governance is still nominally Established with the Church of England, it is today secular.
(There was no other Christian sect to choose when Christianity came to England, at first converting the late Romans; and then the Angles, etc. Its ancient Greek and Russian Orthodox versions were and remain largely confined to those countries, though I believe both now have congregations in the UK and other nations.)
So "England" as a name and entity has nothing to do with angling for fish, it originated centuries before anyone thought of hooking fish out of water, and it has everything to do with the people known as the Angles!