As Tolkien once told us, anyone whose family ever had a claim to a throne will remember it for literally thousands of years. Therefore, given that I live in a country with multiple disowned royal lines, I thought it might be interesting to take a look at some of the alternate claims that might spring up.
Line of Alfred the Great
Alfred the Great (he of the burnt cakes) is generally taken to be the first King of England. His grandson Aethelstan was the first to claim that title, but who's heard of Aethelstan? Nobody, so we'll say Alfred. It all works out the same anyway.
The line down from Alfred was broken a couple of times by the Vikings, but managed to make it through nearly to the Norman Conquest in 1066. Harold Godwinson (he of the arrow in the eye) was not the closest heir to Edward the Confessor - that honour went to Edward the Exile, the Confessor's nephew.
Obviously, the Conquest broke the chain of Saxon kings - but it's possible to trace their line quite a lot further. Edward the Exile's son died without children, but his daughter Margaret married... King Malcolm of Scotland! The Saxon line descends in lockstep with the Scottish royal line from this point, until at last it returns to England in the person of James VI & I (who nearly got blown up by Guy Fawkes). So that's nice. ^_^
Line of William the Conqueror
You might expect that, in all the various wars during the Middle Ages, the 'proper' line of inheritance from William the Conqueror would have been broken, and you'd be right - but it always managed to come back together. Partly that comes from disputed claimants doing away with the rightful heirs (looking at you, Prince John), but the biggest break was the Wars of the Roses, which sprang out of a dodgy assumption of the throne by Henry IV. Henry was a son of Edward III's third son, John of Gaunt, and took the throne from Richard II, son of Edward's eldest - but Edward's second son, Lionel, had an heir of his own.
That heir was Philippa, Countess of Ulster, and after a few generations her heir returned to the throne as Edward IV. His heir was Edward V, one of the Princes in the Tower, who died as a child, leaving his next heir as Elizabeth of York. The selfsame Elizabeth brought an end to the War of the Roses by marrying Henry Tudor (who had a dodgy claim through John of Gaunt, but mostly through defeating Richard III), and the rightful queen Elizabeth became mother to Henry VIII.
(And, a few generations later and by a different branch, the great-great-grandmother of James VI & I again. It's nice when things come together.)
Line of James VI & I
This is the one point where the English royal line gets well and truly broken: James' grandson, James II, was deposed by his son-in-law, William of Orange. The two sides squabbled for a bit (see: the Jacobite Risings), but hilariously enough, after a couple of generations every single branch of James II's line died out. That meant the claim got pushed back up the tree.
It went initially to the children of Charles I, James II's father. Ironically enough, his grandchildren included William of Orange, so he was sort of retroactively an heir to the throne. Other than that, his line almost died out. That would have bumped the succession back another generation, up to Elizabeth Stuart, daughter of James VI & I, whose grandson became King George I... but Charles' youngest daughter, Henrietta, had children, and one of those - just one - managed to have children of her own.
One of those children was Charles Emmanuel III, King of Sardinia, and it is through him that the 'rightful' line of both Alfred and William continues (assuming one doesn't accept the law that excludes Catholics from the succession, which was the principle on which Sophia, Electress of Hanover - mother of George I - was chosen as the foundation of the current line). The current heir would be Franz, Duke of Bavaria; oddly enough, the second in line to the throne after his son is Sophie, Princess of Liechtenstein. Liechtensteeeeeeein! [Shakes fist]
Line of Sophia of Hanover
... is a very boring line, because it's the one we've still got. But, recently the laws in Britain were changed to allow elder daughters to sit higher in the line of succession than their brothers. What would have happened if that rule had been put in place when Sophia was created heir presumptive to the crown of Britain?
George I was Sophia's eldest child, and George II his. George II's eldest was Frederick, who predeceased him leaving a son, George III - but George was not Frederick's eldest child. That honour, and our hypothetical crown, goes to Princess Augusta, Duchess of Brunswick.
Her third child wound up marrying our George IV, but forget her - she had an elder daughter, another Duchess Augusta. Her son was Prince William of Württemberg; his eldest Marie had no children, but his second daughter, Princess Sophie of Württemberg, married King William III of the Netherlands!
... and then all three of their children died before William did. At that point the Netherlands actually adopted the cognatic primogeniture we're discussing here, but passed the throne to the daughter of William's second wife. We need instead to go back up to William of Württemberg, whose third child was another daughter, Princess Katherine. She was the mother of King William II of Württemberg; his eldest was Princess Pauline, who a) was West German director of the Red Cross for many years, and b) was indicted after WWII for concealing Nazis. So, y'know, not great.
Her son Hermann died in 1941, a couple of decaded before her, but left behind a son Friedrich Wilhelm. He died in 2000, leaving behind four children; the current hypothetical King of England by this line is Alexander of Wied, whose heir presumptive is his brother Carl's son Maximilian.
Cognatic primogeniture line of WIlliam the Conqueror
This line would initially follow the actual line down to Henry II, son of the Empress Matilda, but then jumps off to Matilda, Duchess of Saxony. Her eldest child's line appears to die out, following which we bump through various German royalty until we hit Agnes of Baden. Her daughter and heir married Ulrich II of Sanneck, and it seems they only had one child: Frederick, first count of Celje.
From there we make a big jump sideways: Frederick's great-granddaughter Anna married King Władysław II of Poland! Her daughter was named Hedwig, who sadly died without children. Back up we go... all the way to Frederick, to now follow his second son, Hermann I. We can follow his line down to Ulrich II of Celje, whose surviving child Elizabeth married Matthias Corvinus of Hungary. Uuuuunfortunately it doesn't look like she had any kids either... back up we go.
Okay: Hermann I's son Hermann had several children. Frederick's line led to Elizabeth. Anna comes next, marrying the Palatine of Hungary. I have to leave Wikipedia to find her children; the eldest appears to be Ladislaus Garai of Hungary. His eldest daughter Ana was engaged to the same Matthias Corvinus we've already met, but died young. Other than her, there don't seem to be any records of what happened to his children, and so the line drops into obscurity there, with Ladislaus' death in 1459.
Here follows the Line of the Usurper Queen
EDIT: A little further... this source claimed Ladislaus/László had four children, and that the fourth, Miklós, had two children: Jozsef and Ilona. The Hungarian wiki suggests that his line likely died out, which sends us back up again... none of the rest of Anna's children seem to have produced lasting lineages, so we're back up to Hermann II. His last child that we need to investigate is Barbara of Cilli, Holy Roman Empress and Queen of Hungary.
Her daughter was Elizabeth of Luxembourg, whose shenanigans after her husband died actually led to László being deposed, just in case you were wondering how close all these people were. Her daughter Anne had her own daughter, Margaret of Thuringia, who married the Elector of Brandenburg and ultimately gave rise to the German Emperors. Y'know, provided you only allow sons to inherit. Which we're not.
We can follow the Electors of Brandenburg down a few generations, to George William in 1640. His eldest was a daughter, and would be succeeded by her own daughter, Louise Elisabeth of Courland. She married the Landgrave of Hesse-Homburg; their first two children didn't produce lasting lines, and the next five died without issue, but their eighth(!) married the prince of Nassau-Siegen. His eldest daughter's line died out, but his male heir Frederick William had five daughters with a lady named Polyxena.
The eldest of those daughters married Charles Paul Ernest, and they had at least six children. Their second son inherited his father's titles, and their youngest daughter was a writer, but for now, Sophie Charlotte's death in 1759 is as far as I can trace the line of William the Conqueror. o.O
EDIT2: And we're back... Charles & Sophie's first child was named Anna Polixena and died in 1799, apparently childless. She would be succeeded by her sister Eleanore Augusta Amalie, who died 1827, leaving behind her son Ernst Casimir. His eldest was Adelheid, who died childless, leaving Ernst Casimir II to take the throne. Bruno was hie heir, and on his death in 1906 was followed by Princess Hedwig. She married Prince Botho of Stolberg-Rossla, and had Princess Elisabeth, who married the Duke of Mecklenburg, and then... another Duke of Mecklenburg.
She died in 1969, apparently childless. Her brother Cristoph died in 1949, but left behind three children - the oldest of whom, Caroline Christine Hedwig Marie Hermine Luise Friederike Stolberg-Roßla, apears to be still alive! All hail Her Majesty, Queen Caroline-Christine of the United Kingdom!
Of course, she appears to be 107, so perhaps we should transfer our alliegance to her daughter Ruth. Who's eleven years older than Prince Charles... I'm sure it'll be fine. ^_^;
And that, at last, is the end of the line.
INTERSTITIAL EDIT 3: Oh no! Herman 2's second son had kids! All of the below are (probably) vile pretenders! Renounce your allegiance to the Usurper Queen Caroline-Christine; our true monarch waits to be discovered!