As an Irish person, in normal speech the word "scheme" has exactly the same shady connotations as it does for Americans. Calling someone a "schemer" is a common insult. I've always assumed the government started using the word in a rare moment of honesty and it stuck.

In India too, discounts and promotional policies are commonly called 'schemes.' I learned the hard way that in the US, the word has a negative connotation when I asked my rental office about any 'schemes,' they looked at me with total shock.

Growing up in the UK, we would be sent to a “play scheme” during the school holidays. Weird phrase.