The correct spelling is programmed. This is true for both American English and British English.
In the world of computers and IT, the correct spelling of the infinitive form of the verb as well as the noun is always program.
In American English, the correct spelling in every context is program, but in British English, programme is used for all meanings apart from computing (e.g., “Don’t forget to buy your souvenir programme at the concert”). This is true for both noun and verb.
You can use the selectable regional settings in the QuillBot Grammar Checker to make sure you have spelled program or programme correctly.