Hello! I can give a basic response, but I don't have my books with me so I'll have to update later if I can find the time. This story definitely fits within the realm of possibility.

The Ottoman army went through a ton of reforms in the late 19th century. In the 1860s or 1870s they developed a recruiting system that mostly stayed in place until the end of the empire. Recruits were called up locally and organized into essentially regional regiments. After a few years of active duty, soldiers were semi-retired into a reserve system. The Ottomans had several armies, all based in an area of the empire and responsible for the security of that area, although in wartime they could send their troops wherever needed. The Ottoman 5th Army was the one in Syria and Palestine for most of that period, and was based out of Damascus.

"Local" detachments could get sent quite far from the area that the conscripts were raised, and the distance they traveled increased as technology and organization improved over the course of the 19th and early 20th centuries. For example, recruits from the 5th Army were sent to the Balkans in the 1877-8 Russo-Ottoman War and to Egypt during the Urabi Revolt in 1879-1882. Likewise, troops from other parts of the empire came to Syria, as in the case of troops from Albania showing up to put down several Druze revolts in the 1890s and early 1900s. So Ottoman recruits went all over the place.

Now, on to your grandfather. Where would Ottoman conscripts be sent 20 years or so after 1890 (so around 1910-11) and be in a place where they could end up under British medical care? The main culprit would be the war in Yemen. From 1906-1911, the Ottomans fought a Shiite uprising in the hills of Yemen. It was basically their Vietnam. They sent loads of troops into a terrible guerrilla war of attrition. There are still folk songs in Turkish today about sending your sons to Yemen. It SEEMS far to go to Yemen from Palestine, but the Suez Canal was open and the Hejaz and Yemen were the next two Ottoman provinces south of the canal. Syria was the closest major Ottoman province to the Red Sea, and steamship service was common and cheap by that era.

How does this connect with Britain? The British had a colony in Aden on the south coast of Yemen which they operated as a coaling port for their empire. The British and Ottomans got along fairly well in Yemen, and in fact the border they set between Ottoman and British Yemen was basically the line between divided Yemen in the Cold War (another fun fact- those Shiite rebels were basically the ancestors of today's Houthis, and fighting a similar fight versus the Ottomans as the Houthis are today against Saudi and the UAE. That fight is basically the Saudi and Emirati Vietnam, too!). Your grandfather was injured severely enough to be in a coma. It is not unreasonable to think that the British, who had far superior medical equipment in the area, offered to take him in for the Ottomans, and eventually shipped him to India, which presumably had far greater medical capacities than their colony at Yemen. Again, Yemen to India SEEMS far, but west India was basically the next stop via steamship from British Aden.

Another darkhorse candidate would be in some minor conflict in the Trucial States (now the UAE), Qatar, or Kuwait. The Ottomans and British contested the control of those areas (this was before oil, so they were mostly contesting to control shipping routes in the Persian Gulf) throughout the latter 19th century. Both also fought infrequent skirmishes with the local inhabitants. The Anglo-Ottoman convention that ultimately defined the border (and a lot of the modern Gulf borders, for that matter!) was negotiated between 1911 and 1913. It was peaceful, but who knows! This is a fairly understudied area of Ottoman history, but it is entirely possible Ottoman army detachments were in the area to keep the peace. It is more likely they would have come out of the 6th Army, based in Baghdad, but if your Grandfather ended up conscripted and working in the boiler room of a ship or something, its not out of the question! That area is even closer to India than Aden, and any British detachments would have had even more rudimentary medical facilities than even a small colony like Aden, so India would be the closest place for them to send him.

I hope this helps! Hopefully someone who really knows their stuff on the Arabian Peninsula in the early 20th century can chime in!