Various diseases were proposed as a root cause Tiny Tim's condition but let's focus on two of the conditions that plagued kids from lower classes- TB and rickets. It isn't said that he was cured, but that he survived next year- interesting choince of wording.

Rickets- severe vitamin D defiviency that lead to decrease of bone density resulting in various malformations - was caused by poor diet and lack of sun exposure, often starting in infancy due to vitamin D deficit in pregnant and breastfeeding mothers. Dairy and meat were a premium food for the poorest strata of city dwellers and judging from family's reaction to small goose they could rarely afford a proper meat. While rickets affected all parts of skeleton one of the earliest manifestation were bowig femours, corrected with metal braces

Better diet and moving a patients into better dwellings put a halt on development of rachitis florida- the most severe form. The second half of the century saw a rose of prevention programs lead either by stare or charities- famous cod liver suplementation in school and early form of summer camps.

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TB seems to have solid basis- Dickens sister and nephew met their early demise due to "consumption".

TB was one of the most common diseases in the cities during industrial evolution- estimations that half of the London's population was infected are on the lower side. While pulmonary tuberculosis was the most common manifestation the other forms were also prevelent with bone tuberculosis (Pott's diseases) disproportionely affecting pediatric population. While the causes of TB were poory understood and antibiotic treatemnt weren't to be discovered till 1940s it was widely understood that TB thrieve on malnoutrishment, pollution and poverty- treatment involved high calorie, rich in meat and dairy diet and moving to countryside in regions understood as a healthy climate. That in some cases could result in spontaneus remission and even if the patrient wasn't cured better living condition resulted in prolonged living.

Other proposed ilnesses- polio, children paralysis- aren't necessairly fatal but contibute to lifelong suspectibility to infective diseases that could be amplified by malnutrition, polluted air and lack of physical excesise. It's easy to see already fragile boy to fall prey to seasonal infective disease.

TL;Dr Whatever condition Tiny Tim had the bad living conditions weakend him further and made him prone to infections. The effects of living conditions of the children were understood and doctors consulting Tiny Tim could give sound advice on diet and living conditions that could be implemented with help of wealthy partron.

So better payment for father, better living conditions and Scrooge funding "countryside cure" could improve his health and prevent early death even without full remission

Further reading https://jamanetwork.com/journals/jamapediatrics/fullarticle/1107722