In England, particularly in London, if we have nobody with us we may say 'I am on my tod'

How is this saying related to an American sportsman who was famous when working in America before becoming famous here when working in England ?

Here you are darling, it's slang for Tod Sloan, an American jockey:

Urban dictionary

and from Wikipedia:

"The name of Tod Sloan left a mark on the English language. His fame spread to London where the name was adopted into the rhyming slang to mean 'own' as in 'on his own'. Hence, someone 'on his tod' is alone. This adoption of a rhyming phrase then the dropping of the word which rhymes is normal."

Indeed. He was the most famous jockey in America. He came to England about 1895. He was instantly notable because of his idiosyncratic style of riding. He rode 'very short' (a reference to the length of his stirrup leathers, not his height )with his knees tucked up, as all jockeys now do today.Back then they rode long, as though riding in a gymkhana or at a hunt. This style marked him out and the public soon took to him with great, if initially amused, affection.

So he is immortalised by his name living on in everyday speech, though he himself is long forgotten. He joins the few who are so remembered but now otherwise lost to public memory: an (American) example is Gordon Bennett, whose name has long been a euphemism for an obvious blasphemy and another is the poor Fanny Adams, the victim of a child murderer who dismembered her body. She was 'sweet Fanny Adams' or 'sweet F A', whose name now means 'very little' or 'almost nothing', that being what was left of her.(Sailors, with a gruesome sense of humour, had first applied her name to tinned minced meat issued as rations).

I keep a contemporary cartoon of Tod Sloan here by the computer desk : appropriate as this is Newmarket, the British Headquarters of Racing, and because at the computer I am 'on my tod'