Great Britain limits provision of military intelligence to the US, abrogating the Five-Eyes agreement. I don't see Canada, New Zealand and Australia being too upset with this immediately.
We could also restrict entry to the various US staffed intelligence bases in the UK (Menwith Hill, Croughton, MoD) and mutter things about flying rights treaties at places like Diego Garcia, Ascension Island and the numerous UK bases. Perhaps it's also time to look at the need for submarine basing at Holy Loch and Faslane?
We can follow this up with the threat of unplugging all UK intelligence sources from NATO after the Falklands Conflict is over. We need whatever they can get while the fight is on, and we did get a lot of help from all corners. NATO as a whole would be rather upset at the loss of UK provided intelligence.
Other European countries with overseas territories (France, Spain, Portugal etc) will have their own nervous twitches at the thought of the United States not supporting them if they had the same issue as in the South Atlantic.
At this point every NATO country will be looking askew at the US Government and wondering if impeaching Reagan is possible...