The Oxford English Dictionary traces sugar as slang for money as far back as 1862, in an article in Cornhill Magazine about thieves' cant. I found the article in Google Books.
We have just touched for a rattling stake of sugar at Brum
is translated as the thief having robbed someone in Birmingham for a large amount of money. The paragraph above suggests an analogy with sweet to refer to an unsuspecting (and therefore good) target of crime. Now, sweet has been used in a general sense of "something desirable" for a very long time. OED has it as "an emotional epithet expressive of the speaker's personal feelings as to the attractiveness of the object" from 1779, and as "yielding pleasure or enjoyment to the mind or feelings" since 888 (!).
As for heavy, its metaphoric usage as something serious or important is also very old, certainly going back over a thousand years.
Putting it together, heavy sugar is a serious amount of money; and it's "sugar" because money is a sweet thing to have.