Let's be clear: the UK does have a Constitution, it is simply distributed, not contained in or confined to a single document. None of the legal reasoning surrounding the actions of government or Parliament in this particular case, however, involve anything more than the most straightforward questions; there is no crisis.
While I believe that in the case of the US federal government, there would be more scope than in the UK to proceed with negotiations and afterward present the results to Congress, as a general rule for states where there is specific legislation affected or nullified by larger executive efforts, then it is inevitably and only the legislature that can authorize executive action by removing or modifying that legislation. This is even the case in most dictatorships that I know of, with the legislative process being no more than a pretense but existing nonetheless.
To simplify what I've been speaking of, partly using as reference American negotiations on the nuclear agreement with Iran:
1. Legislature passes resolution of approval for terms of engagement.
2. Officials of government discuss with foreign parties or under auspices of foreign/multinational body.
3. Officials of government present terms for finalization.
4a. Inconclusive, return to Step 2.
4b. Legislature passes resolution of disapproval, possibly cancelling the enterprise in its current form.
4c. Legislature passes resolution of approval, finalizing terms.
If Step 4c is taken, then if course there must be further legislative changes to fit whatever the results of Step 2 were fit into the existing legal structure.
In the specific case of the United States, partly due to the nature of the federal executive and the highly multinational nature of the topic, Congress has passed some legislation pertaining to things like waivers for sanctions in the case of following the roadmap, which was agreed to by Iran and the body involved, but Congress AFAIK never passed either approval or disapproval of the final agreement as it stands.
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