graft (the horticultural concept) derives ultimately from the Greek for "stylus", after the knife used in the process, and this etymology is considered reliable.
graft (British English for your job) is not well understood but speculated to come from the word graft (meaning ditch/trench), possibly from Dutch.
graft (American English for corruption) is speculated to come from the British word for your job. It's not felt to be related to the horticultural term.
At least in Australia both meanings are common and it has always seemed pretty clear from the context which meaning was which.
This may be another case of English speaking communities being separated by the same language.
It is understood elsewhere in the UK but not used routinely. It might, say, be deployed for additional effect by someone not from northern England and imply very hard work. However, it might be pronounced in an exaggerated, imitation "generic Yorkshire" accent in which case it gets a bit complicated to explain.
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Actually, isn't the US word "grift" rather than "graft"?
"Grit" is a perfectly valid en_GB word meaning the same (enduring hardship.)
A lot of the more unusual spellings in British English (in particular) relate to trying to distinguish words by their origin.
Spoilers: US usage is always negative, Australian usage mostly matches US, British is mostly positive, India is both. Someone asks the same spelling question, and gets the answer that it is because the word was only written down in either meaning quite late.