The simple and sad fact is we have no real idea, except it appears to have been something to do with his lungs, leading to conclusions of the effect of the cold weather, influenza, and pneumonia.
However, 21st century humans know that cold weather has little causative effect on flu and pneumonia, being viral and, in the case of pneumonia, possibly bacterial or fungal as well, but trying to cope with what to Yemmerrawanyea would have been the extreme and completely unknown cold of a London winter would almost certainly have weakened his system.
Although Yemmerrawanyea died in May, 1794, he first appears to have begun developing illness in September, 1793. Bennelong already appears to have been ill, but by September he was recovering.
The records related to Yemmerrawanyea's illness show the use of treatments relevant to a leg infection. How it developed is unknown, although some historians speculate he might have cut his leg when on a swimming trip in August.
An alternative is that Yemmerrawanyea developed tuberculosis (TB). TB is a bacterial disease passed on by air-borne sputum.
While the precise cause of TB was not known in 1794, the existence of what was at that time a sub-microscopic cause (that is, it couldn’t be seen, even with a microscope) was certainly theorised, although how it spread was unknown. There are several different types of this insidious disease, and, indeed, several similar diseases.
If Yemmerrawanyea's illness was pulmonary tuberculosis, which, if he had it at all, was the most likely culprit, it infects the lungs, and can cause skin-breaking sores, or "ulcers", at the ends of long bones, like leg bones, and at ankle and knee joints.
These sores can fill and swell with pus. Other symptoms can be a cough with sputum, which may get worse and worse and contain blood as time goes on, and fits of substantial sweating.
The major argument against this belief is that TB was exceptionally common in 1794, although extra-pulmonary TB, the sort with sores and so on, was comparatively rare.
Nonetheless, it's highly probable a doctor with Blane's experience would have seen quite a few of these cases and would surely have recognised it for what it was.
However, we don't know what Blane was treating because we don't have his notes, if he had notes. The only information we have are the records of amounts spent and what they were spent on.
This data only gives us a hint about the symptoms the poor young man was suffering, and no information, apart from guesses, on the overall illness or illnesses causing those symptoms. Nonetheless, the symptoms appear to be roughly consistent with TB, rather than a string of unconnected illnesses.
While these anti-vaccinationists, thinking that a smallpox vaccine made out of cowpox material would cause them to grow cow parts, are incredibly stupid, one can understand their suspicion of doctors of the day who could sometimes identify an illness, but despite much salesmanship they could very rarely cure it. And even when they did, it was usually an accident. This is what faced poor Yemmerrawanyea.
These are modern TB sufferers. If he had TB, this may well have been what Yemmerrawanyea looked like towards the end of his life.
These are the kind of sores, or "ulcers", Yemmerrawanyea could have had. Mind you, they could be a heck of a lot worse, but we didn't want to make you too ill.