Through Iran and Turkey.British could do so in 19th century,but why? canal in Egypt was better.Ships always take more then trains.
Russia try that,becouse they could be cut off from seas - no danger for England.
I was thinking more along the lines of Germany aiming to do this, but if Germany will become a British ally, then this problem will become less severe.
And you also need to cross either Afghanistan or British India after Persia to get to China.
The Arabic numerals didn't exist yet, so that would be a bit of a problem. 'Zero' did exist, but was only used as a place-holder. To use it was you'd want it to be used, it needs to develop into a number in its own right, so it can be employed usefully in calculus. This only happened somewhere between AD 400 and AD 500 in OTL, so some developments would be needed there, as well.
Britain actively tried to create the Cape-to-Cairo railroad, and although far more nebulous, the notion of Cape-to-Singapore was certainly there. This would allow the British overland route to largely encirle the Indian Ocean.
Indochina is actually the major obstacle, since (as we see in OTL) the least impractical thing to do is to cut west from Southern Burma to Bangkok, then Saigon, and then along the Vietnamese coast. This is a huge diversion, in terms of overall distance, but it's more practical than trying an inland route across the mountains.
Realistically speaking, Cape-to-Singapore has merit, but from Singapore onward, it's more practical to just go by ship under any given circumstance.
The major obstacle for Cape-to-Singapore is Persia. I think that any ATL scenario that has Britain going for some workable version of Imperial federation and winning (or, possibly, averting) the Great War can quite realistically get you this result. Southern Persia could realistically be turned into a perment sphere of influence, absent the huge losses of the war, and then the rail project could be constructed over a few decades. I think that by the 1940s, you'd realistically have Cape-to-Singapore.
Who gets French Indochina in the event that of a German-British-Japanese coalition defeating the French in an alt-WWI?