'Programme' is the British spelling for the noun 'program'. However, the 'program' spelling is the generally accepted form when referring to computer programs.
As a verb, the situation is slightly unclear. My 1995 Oxford dictionary lists 'program' as both a noun ("series of coded instructions for a computer") and a verb ("instruct a computer by means of (a program)"). However, 'programme' is listed only as a noun.
Although several web pages contradict this (stating that 'programme' can be a verb), I trust my dictionary more. If 'program' became verbed as a result of its use in computing, this would make the case even stronger.
At any rate, I would....
- Watch a television programme
- Write a computer program, and
- Program my computer or video ('Programme my video' just looks strange...)
Dictionary: Oxford Quick Reference Dictionary & Thesaurus
Also see: http://www.oup.co.uk/
This was originally intended to be a brief, informative node, not a bloody linguistic investigation.