In British English, the spelling program is used when referring to computer programs, but the spelling programme is used for all other purposes.
The following entry is taken from a reputable British dictionary1
(I have omitted the entries relating to non-computer meanings for the word):
programme or (N Amer) program
5. (usually program) computing a set of coded instructions to a computer for the performance of a task or a series of operations, written in any of various programming languages.
verb (programmed, programming; N Amer also programed, programing)
3. to set (a computer) by program to perform a set of operations.
4. to prepare a program for a computer.
Note that this British dictionary uses the spelling program for all uses related to computing.