I wrote a rough timeline on this idea and this was the section on Charlotte's son - like others before, I had Charlotte and Leopold only having a single child...
Name: Francis George Augustus
Reign: 1878 - 1885
Life: 1817 - 1885
Consort: Princess Alexandrine of Baden
Issue: (1) Prince Leopold Ernest Francis
Their Deal:
George V was born under otherwise traumatic circumstances and very nearly died the night of his birth - but he survived and was brought up by parents who doted upon him. His grandfather, George IV, had never been the most attentive father and this didn't change with his grandsons birth. As such he was much more influenced by his fathers side of the family and his father specifically. At the age of thirteen, his father Leopold was crowned King of the Belgians and the heir to the British throne also became the heir to the Belgian throne. It would be a long time until he attained either - his father died in 1865 and his mother not until 1878. King of the Belgians at 48, King of the United Kingdom at 61 and dead by 68.
He was officially created as Prince of Wales shortly before he turned 21, as part of his mothers birthday celebrations.
George V often felt that he might as well have been the spare rather than the heir - his life was fifty years of waiting for his parent to die so that he could rule. Nonetheless he did his duty and married Princess Alexandrine of Baden who had previously been destined to marry his cousin, Ernest of Saxe-Coburg-Saalfield and Gotha, and produced an heir (but only one) of his own: Prince Leopold Ernest Francis, named after his father, maternal and paternal grandfathers (Francis, Dukr of Saxe-Coburg-Saalfield and Gotha and Leopold, Grand Duke of Baden) and his godfather (the aforementioned Ernest of Saxe-Coburg-Saalfield and Gotha).
The influence of his fathers family expanded to his political leanings - much like his father, he encouraged economic modernisation. After the creation of Belgium's first railway in 1835, Prince Francis (as he was then, and at the age of only 18) petitioned his mother to press her government into encouraging railway development and reform in the United Kingdom. His friendship with married mathematician and computer pioneer Ada Lovelace led to questions about her marital fidelity - and the Prince's own as he had married in 1842. He also attended the Royal Institution Christmas Lectures for many years - from 1835 until his death with only a rare absence. After the first - the 1835 lecture in which Michael Faraday discussed electricity - he arranged a private meeting with Faraday and remained a supporter of the scientist, writing to him for the next 32 years until the scientists death in 1867.
He enjoyed a similar friendship with Isambard Kingdom Brunel and attended the launch of the SS Great Western in July 1837. The Prince had everything going for him with regards to becoming a progressive and respected monarch. And after being King of Belgium prior to the death of his mother and his accession, he was coming to the role with thirteen years worth of experience of being monarch and interacting with parliament. Early signs were that King George V would have continued his programmes of modernisation - he championed the construction of Tower Bridge (although construction did not begin until after his death) and the Mersey Railway Tunnel (not opened until after his death) and helped found the Royal College of Music, opening it in 1883 with his wife, Queen Alexandrine.
His influence carried further than encouraging economic and technical advancement. He was offered numerous trading colonies of the Congo Basin as his personal property outside of the control of the Belgian government and its finances. He refused and the colonies were officially ceded to the Belgian government and operated as colonies of the state. That was only his first move. His next came after taking part in the first state visit of a ruling British monarch to the United States in 1876 as part of their centennial celebrations - although the visit itself caused some tension, it led to a correspondence between King George and the American Presidents (Ulysses S. Grant at the time of his visit, subsequently Rutherford B. Hayes, briefly James A. Garfield and finally Chester A. Arthur) which continued through their respective successors - modern political historians consider this state visit as the start of the 'special relationship' between the two countries. Having seen how the slave trade had fractured an entire country, he determined to take it upon himself to almost single-handedly block the expansion of the European powers into Afria.
At the Berlin Conference in the Prussian capital in late 1884, the King had planned to force through a gentleman's agreement between the leaders of the major European powers for them all to subscribe to a plan that, outside of the establishment of limited trading colonies such as those he had previously ceded to the Belgian government, they would not seek to occupy or further colonise the African continent. However, King George did not attend the conference (but he was represented by his son, the future King Ernest) as he had been diagnosed with inoperable lung cancer shortly before he had been due to set off and died less than ten months later on 13th July 1885. His funeral took place two weeks later. A funeral procession through London ended up with him being buried at St George's Chapel in Windsor Castle. He was survived by wife, Queen Alexandrine, who was styled as Queen Alexandrine, the Queen Mother through the reign of her son, King Ernest, until her death on 20th September 1904.
Succeeded By: King Ernest