State testing unarmed ballistic weapons off some waters abroad?
<blockquote>
  <p>The Sunday Times reported an unarmed missile had been set off from the
  submarine off the coast of Florida but, rather than head towards
  Africa, had veered towards the US.</p>
  
  <p>[ BBC News, <a href="http://www.bbc.com/news/uk-38714047" rel="nofollow noreferrer"><em>Trident: Defence Secretary refuses to give test missile
  details</em></a>, 23 January 2017 ]</p>
</blockquote>

<p>The missile is a <a href="https://en.wikipedia.org/wiki/Trident_(missile)" rel="nofollow noreferrer"><em>Trident</em></a> SLBM. It was fired by the <a href="https://en.wikipedia.org/wiki/HMS_Vengeance_(S31)" rel="nofollow noreferrer"><em>HMS Vengeance</em></a> from the <em>Royal Navy</em> (Britain). In the preamble to the <em>Treaty on the Non-Proliferation of Nuclear Weapons</em>, for which Britain was a <a href="https://en.wikipedia.org/wiki/List_of_parties_to_the_Nuclear_Non-Proliferation_Treaty" rel="nofollow noreferrer">depositary state</a>, I read the following:</p>

<blockquote>
  <p>Desiring to further the easing of international tension and the
  strengthening of trust between States in order to facilitate the
  cessation of the manufacture of nuclear weapons, the liquidation of
  all their existing stockpiles, and the elimination from national
  arsenals of nuclear weapons <em>and the means of their delivery</em> pursuant
  to a Treaty on general and complete disarmament under strict and
  effective international control,</p>
  
  <p>Recalling that, in accordance with the Charter of the United Nations,
  States must refrain in their international relations from the threat
  or use of force against the territorial integrity or political
  independence of any State, or in any other manner inconsistent with
  the Purposes of the United Nations, and that the establishment and
  maintenance of international peace and security are to be promoted
  with the least diversion for armaments of the worlds human and
  economic resources,</p>
  
  <p>[ From the preamble of the <a href="https://en.wikisource.org/wiki/Nuclear_Non-Proliferation_Treaty" rel="nofollow noreferrer"><em>Treaty on the Non-Proliferation of Nuclear
  Weapons</a> (NPT)</em> (1968) ]</p>
</blockquote>

<p>There are also a series of international legal instruments such as <a href="https://en.wikipedia.org/wiki/Nuclear_weapons_testing#Treaties_against_testing" rel="nofollow noreferrer">treaties</a> which prohibit the formal testing of nuclear weapons which <a href="https://en.wikipedia.org/wiki/Nuclear_weapons_testing" rel="nofollow noreferrer"><em>Wikipedia</em></a> introduces as the "experiments carried out to determine the effectiveness, yield, and explosive capability of nuclear weapons".</p>

<hr>

<ul>
<li>Would a country be technically conducting a <em>nuclear weapons test</em> according to any international legal instruments by doing what Britain was reported as doing?</li>
<li>Irrespective of whether such doings were formally a nuclear
weapons test or not, would such a test infringe on any rule of
international law?

<ul>
<li>In particular, are there any rules of international law allowing or prohibiting a country from so testing <em>ballistic missiles</em>? Do the nuclear weapons states recognized by the non-proliferation treaty (NPT) enjoy a specific status in that respect?</li>
<li>Would being in international waters as opposed to within 200 sea miles of a country make any difference (I do not know where the missile was deployed from exactly)?</li>
</ul></li>
</ul>

<p>First of all, the wording in your question hints at the UK deliberately launching a missile (even if it is a test one) towards the USA. Nothing in the news piece you link to support that supposition, the general idea is a missile that was launched towards the Atlantic Ocean that steered off route.</p>

<ol>
<li><p>From your link:</p>

<blockquote>
  <p>The treaty recognizes five states as nuclear-weapon states.</p>
</blockquote>

<p>The NPT limits <strong>other states</strong> from researching (including testing) nuclear weapons and/or delivery vehicles. In exchange, they get access to nuclear technology for civilian purposes.</p>

<p>Treaty preambles are general, introductory texts that explain the intent of the treaty. They are not binding, and the language clearly shows it (Hint: It states "Desiring to", "Recalling to". Binding agreements use "Shall", "Shall not", "May" or "May not", and are more specific about what are the restrictions).</p></li>
<li><p>As stated above, the test was not "in North-America". It was from international waters "towards Africa", probably to be sunk before it reached land<sup>1</sup>. There is no notion anywhere that there was an intention of violating the airspace of any country.</p>

<p>Certainly, the UK would be responsable if one of their missiles test fails and the missile ends causing damages, and the USA can protest any violation of its airspace, even if unintentional. But it is not "a test of ballistic missiles in North America", and certainly the USA has no more rights outside its territorial waters than, say, Gambia does.</p>

<p>The only restriction would be the internationally stablished procedures for testing missiles, which are setup to avoid any kind of issues (the test scheduled date and path is published<sup>2</sup> with time enough for third parties to take notice).</p></li>
</ol>

<p>So yes, it is <a href="http://www.bbc.com/news/uk-scotland-scotland-politics-38716446" rel="nofollow noreferrer">business as usual</a>.</p>

<hr/>

<p><sup>1</sup>Idea: African countries are countries, too, and they have the same right as the USA for their airspace not to be violated by the military from other countries.</p>

<p><sup>2</sup>Both to allow for air traffic to avoid dangerous areas, and to ensure that anybody detecting the launch does not think that it is an attack.</p>

international weapons treaty law-of-the-sea