This is not a question to which the answer is immediately obvious. That is because if anybody had reliable information about this subject reinforced by evidence, then the matter would be passed to the world's intelligence agencies; the White House; the UN Security Council; and so on and so forth. Instead all we have is speculation in the media that refers to intelligence sources (particularly British ones) but without identifying the sources of their information. This illustrates an intriguing change in the War in Ukraine, compared to previous wars: intelligence agencies, normally highly secretive, are revealing their conclusions to journalists. Is this a good thing? Probably not; it provides information to the public, unfamiliar with the methods of the world's intelligence agencies and hence unable to scrutinise their conclusions, and this may cause undue panic.
The popular assumption is that intelligence agencies, being experts in their field, must be right and are highly reliable. But that is not true; the opposite is the case. Very roughly, intelligence agencies pick up pieces of information from a single source and then write them into a report that is reviewed by various bureaucrats more senior in the hierarchy. Those bureaucrats also look for corroborating sources, often cutting off in the course of their search for corroboration the original informant who it is feared might have an incentive to fabricate in conjunction with the collaborative sources to prove his or her value. Then, when a corroborative source is found, the intelligence agencies typically increase the probative value of the original information. With sufficient different pieces of corroboration, the piece of intelligence is given a ranking (e.g. highly likely; unsafe, etcetera).
The problem with this entire structure, and the problem remains irrespective of whether one is dealing with human intelligence or IT intelligence (i.e. hacking persons' electronic accounts), is that because of the secrecy by which intelligence data is conveyed without making proper enquiries of the person providing it as to its veracity, is that several sources of approximately similar information, presumed to be corroborative, may not be corroborative at all if all the sources have the ultimate same source. To understand why this is, consider the structure of an upside down dotted triangle. The final intelligence sources are the dots at the top: the original common source is the original dot at the bottom; and because intelligence collection is secretive and does not permit the evaluators of the information to establish whether it has just one ultimate source or several.
The Russians are masters in manipulating this structure of intelligence gathering, passing on to multiple sources substantially similar information; ensuring the sources have no way of communicating with one another; and then presenting their adversaries with a highly complex fait accompli which appears to be corroborated intelligence from a number of sources but in fact it all derives from a single piece of disinformation intended to deceive the enemy or solidify their version of the facts on the ground.The Iranians are also ingenious at this, using huge long lists of people all over the place to sew maximum confusion in the disinformation they are planting.
Of course intelligence agencies have tools to mitigate such effects (e.g. comparing human and IT intelligence); but from a lawyer's perspective none of them are as satisfactory as the lawyer's ingrained skill of cross-examination. It is very difficult effectively to lie under cross-examination, because while you can make up some associated details as a witness, you can't make them all up. So cross-examination, undertaken effectively, involves finding the seemingly irrelevant underlying facts that the witness forget to make up. The only effective way of doing this, in the face of a highly effective cross-examiner, is to say the truth that is very easy not to make up.
Let us return to the initial question, that is to ask whether British intelligence that Moscow is planning to use thermonuclear warheads in the invasion of Ukraine. Prima facie, this is virtually impossible to assess. It might well be a massive deceit sent by the Kremlin up the paths of our reverse triangle; and we have no way of assessing it. So, with all courtesy to British intelligence, we cannot rely upon their assertions unless they open their files to us for inspection. They will not be doing this.
The purpose of the Russians doing this, if that is what they are doing, is to scare the living daylights out of the western powers currently funding Ukraine and supplying her with arms. If the west stops doing either of these things, the Ukrainians cannot stop the invasion and Russia wins the war. So it is a ghastly piece of brinkmanship combined with an obscure Russian ultimatum, just as was the Cuban Missile Crisis in 1962. The Russians got they wanted out of that brinkmanship: removal of NATO thermonuclear ballistic missiles from Turkey. What the Russians are planning as demands not to use their nuclear arsenal on this case is not yet clear.
Nevertheless the point remains that the Russians are world experts at brinkmanship, gambling huge stakes to obtain a tactical advantage.
So probably what we need to do is remain calm and not circulate this kind of information to the general population. (After all, the Russians aren't; they have imposed a news blackout). All the nuclear options are after all, fairly bad for Russia.
Russia detonates a thermonuclear warhead over a Ukrainian city. The Americans, after hesitation, drop a thermonuclear warhead over an equivalent Russian city.
Russia drops a thermonuclear warhead over RAF Menwithhill in North Yorkshire, England, the US Government's principal detection and spotting centre for intercontinental ballistic missiles. Then either the UK or the US will drop a similar thermonuclear warhead over an equivalent Russian settlement.
Russia drops a thermonuclear warhead over a US city. The United States responds by dropping a thermonuclear warhead on Moscow.
(RAF Menwithhill, North Yorkshire, England)
There are various escalation scenarios from each of these axioms, most of which result in the end of the modern world. (US political sources in the 1960's used to call this 'mutually assured destruction, or 'MAD', and it kept the world safe during the Cold War. It has long been established that there is no such thing as a 'first strike' capacity - an ability to use nuclear weapons to destroy the opponents' nuclear capacity - without their having the opportunity to fire back first. Submarine launched ballistic missiles and the like soon put paid to that.)
The idea that Russia is defending her existential interests and this requires nuclear response is an American geopolitical idea. The Russians are far too callous and ruthless to do that. After all, they're just going to take the Donbass and Ukraine's South the old fashioned way. If America keeps funding this project, US inflation, gasoline prices and public debt will cause the US voters to remove the Democratic Party in Washington DC; and President Trump will soon be back for a second term. Then Putin will do a deal with him - a for more agreeable US bad guy than Joe Biden.
So there won't be any nuclear exchanges; the intelligence is wrong and in all likelihood created by Russian intelligence agencies to deceive, confuse and alarm; but the fact is that neither the United States nor the Russian Federation, the world's two biggest nuclear powers, are going to go to war with one another whether using nuclear warheads or otherwise. It makes no common sense; and intelligence that makes no common sense is usually wrong.
Our cost of living will go up in the West as a result of this war, and governments will fall as a result; but we are not looking at a nuclear holocaust. As every military strategist knows, the point of nuclear weapons is not to fire them; it is to point them at people. We may rest comfortably in our beds.