Is program correct in British English?

Yes, program is correct in British English when it refers to computers or IT. When used as a noun or verb referring to a schedule, list of events, or printed information about an event, the correct spelling is programme.

In American English, it is always spelled program, for all meanings and for both the verb and noun.

The QuillBot Grammar Checker has selectable regional settings, so you can be sure you have the correct spelling of program or programme.