Under English law, any child born to a Royal out of wedlock is ineligible to succeed the throne, which extends to the child's heirs. This can never be reversed, even if the child is legitimated by the Royal parent. Therefore, despite being the "only" surviving member of the Windham line, Ralph could never take the throne since his father was born out of wedlock.
While this movie is fictional, King Ralph and King Cedric would not be titled as Ralph I and Cedric I. They are only numbered as such once there has been a second monarch of the same name. Hence Queen Victoria is not referred to as "Victoria I."
Ralph is repeatedly referred to as the King of England. The Kingdom of England ceased to exist in 1707 during the reign of Queen Anne, following the Act of Union between England and Scotland. Therefore the last King of England would have been King William III. The correct title would be "King of the United Kingdom" (for short) or "King of the United Kingdom and the other Commonwealth realms" (in full).
The Prime Minister and Lord Percival refer to the house of Wyndham (Windsor) and then indicate that the next in line for succession would be descendants of the house of Stuart, of which Lord Percival is one. The next house in line for succession would in fact be the Hanoverians.
In the establishing shot of Buckingham Palace at the beginning of the film, no Royal Standard is seen flying above the palace, which would signify that the monarch is in residence. Immediately following the establishing shot, it is shown that the monarch is, indeed, in residence and thus, the Royal Standard should be flying.
While the film states that the entire British Royal family has been killed and an heir cannot be located, the actual line of succession to the United Kingdom monarchy extends to hundreds of nobles throughout all the nations of Central and Western Europe. An even longer extended list, tracing to links with the royal family as far back as 1800, extends into the thousands. Thus, even if all of the immediate British family were to die, there would be plenty of people left to take their place.