This is a rare case where fridge logic delayed and altered a film after it was believed finished. In the cut of the movie screened for critics (and also the director's cut released on DVD), Stuart is Leopold's great great grandson. Critics noticed a problem with this plot point that the director hadn't. When Kate travels back in time with Leopold, she becomes (and always was) Stuart's great great grandmother. This means Stuart had a long (and apparently sexual) relationship with a relative. The plot point was edited out to avoid this problem which caused the film to miss its announced release date.
While Leopold's decision to marry Kate did nothing to address his family's debt, by the end of the film, both Leopold and Kate know with certainty that his elevator invention will be quite successful and can take aggressive steps to ensure the maximum financial return on his work. In addition, Kate is from the future and they can use her knowledge to guide investments in real estate, businesses, or stocks. They may have some lean years to live through but it seems there's considerable potential for the Mountbatten family to regain their wealth.
Because the inventor of the elevator is brought to modern times using time travel, all elevators start malfunctioning. One of the characters explicitly brings up that if he is not returned quickly, people will vanish that would have been his descendants. Presumably the same thing would happen to them as did happen to the elevators-first they would all start gaining severe medical problems until eventually they all died.
How well is Kate going to get along stuck in the 1800s? Her job is completely impossible there, so she has no way to make a living, which leaves her dependent on Leopold (in a time without much in the way of women's rights), and if their relationship should go bad, she's up the creek without a paddle.
