"Program" is American English. "Programme" is British English. In Australian English, "Program" and "Programme" are both acceptable. I think it's more natural to say, "I programmed the machine to wash the delicates".

Programme is British spelling. The sentence isn't very natural. I'd say something like "I used the 'Delicates' setting (for this load)".

Thank you very much Angela :) I didn't ask about the US because I believe "program" would be used there, but I wasn't sure whether "program" or "programme" would be used in BrE.

Not that you asked, but US: I set the machine on "delicates."