No degree, but I have done a good deal of research on the Falklands War and the build up to conflict and written numerous papers and performed several presentations on the topic over the past few years.

There are several versions to this story, all being reported from sources who were not too keen on the Iron Lady. The Ronald Reagan aspect I think is something you are mistaken on. Reagan and Thatcher had an excellent working relationship, he even went so far as to offer the UK a US Navy Aircraft Carrier for use during the conflict.

The Exocet missile was of French make, and it was the French President Francois Mitterand, whom Thatcher pressured for said codes (codes in this sense meaning homing signal). The reason this strikes a nerve with the French people/president is because Thatcher publicly stated that Anglo-French relations would suffer a "devastating blow" if France did not cooperate over the missile crisis. This made France look like the bad guy who did not want to help end the war, but also openly pressured them to devalue their Exocet technology. However France cooperated fully. In years after members of Thatcher's staff claimed she had threatened nuclear action whilst speaking with the French president (I believe his staff said so as well? unsure of that), but giving the French cooperation, it seems unlikely it ever reached that point. Perhaps it is something she said angrily out of context in the privacy of her staff, but it was never a genuine threat or statement.

Thompson, Julian. No picnic: 3 Commando Brigade in the South Atlantic, 1982. London: L. Cooper in association with Secker & Warburg, 1985.

Hastings, Max, and Simon Jenkins. The battle for the Falklands. New York: Norton, 1983.

Edit: removed the word "disabling" before "codes" because of the implication it could mean "self-destruct codes".

Edit 2: Sources

I had always heard the story, that the SAS did a black bag job infiltrating the Argentine base, and messed up the guidance system software. Heard this from some American Naval officers in 1986.