So here's the first question I have - if the Germans did have these weapons, why weren't they actually used? You did say "between 1 and 3 bombs already in existence" by late 1944, and of course a thermonuclear warhead is extremely powerful - and yet they weren't even used to try and destroy a bridge, let alone a Shock Army?
The hydrogen bomb, which was an SS project conducted to a large extent by Austrian scientists and other personnel,
was not finished prior to war's end so far as is known. The declassified Top Secret British intelligence document cited upthread provides names of some of the major figures involved in its development. For sure, the British got their hands on one of them when T-Force personnel literally knocked on the door of Wilhelm Groth's laboratory and quickly loaded him aboard a plane that flew him directly to the UK.
There is however considerable evidence that at least six (6) prototypes and three (3) operational nuclear weapons were completed by WWII Germany. There may or may not have been additional devices beyond these, but the best available evidence describes what appear to have been
an undetermined but probably not large number of fissile cores aka "fission pits" rather than additional completed bombs.
Again by way of British intelligence, specifically German speaker Edmund Tilley:
Edmund Tilley. Brief Operational Report on [censored] and Other Germans and Italians Connected with Project Abstract. 19 August 1947. [NARA RG 319, Entry A1-134A, Box 29, Folder Operation Oberjoch]
1. On 11 August 1947 [censored] on the Italian phase of PROJECT ABSTRACT, which, in a few previous reports, was called Operation Arrival or Arrivederci, [censored] Headquarters, USAFE,by Captain R.R. SNEIDER of the CIC Detachment, BAD KISSINGEN. Major A J. LEOCHA[censored]
2. [Censored]
SANITIZED COPYSENSITIVE INFORMATION DELETED[...]
New Facts and Re-affirmation of Statements on PROJECT ABSTRACT.
4. Thorough cross-examination has not been effective in shaking [censored] on matters directlyconcerned with or related to PROJECT ABSTRACT.
a. Atomic research and development at TUCHELER HEIDE was coupled with research on guidedmissiles, in 1943 and 1944. (Note: In March 1947 Professor W. von BRAUN admitted that in thesummer of 1943 he had been interested in atomic energy for use in V-2. Von BRAUN claimed to have dropped the project for lack of available raw materials. It may be mere coincidence that [censored]also places the beginning of combined research and development in 1943.—In March and April 1945the undersigned heard rumors in Germany of such a combination. The most persistent rumors in I.G.FARBEN circles had it that this combination would be linked with Chemical Warfare, especially with the new nerve gases, i.e., the TABUN series. These rumors were repeated by responsible members of I.G. FARBEN, who added that this vague plan or hope had been abandoned. At the time no progress was made in the investigation of the atomic side of the problem because all effort was concentrated on a solution of the new Chemical Warfare problem. [Censored] now reveals that the ampullae (phials) he saw in four boxes in Italy had originated with I.G. FARBEN. (See b below).
b. Four boxes, probably originating in the Luftwaffe Ministry in Berlin, were sent in March 1945 bySD, POTSDAM, to ITALY in the care of a FELDWEBEL (Sergeant) HEINZ (surname forgotten!),[censored] and a Private (name forgotten!). The boxes contained (i) reports and research data onV-weapons and atomic research; (ii) 40–50 small ampullae (phials) "full of a whitish liquid", labeledU-234, U-235 and PLU, stamped "IG" (= IG Farbenindustrie); (iii) a new and not completely developed optical instrument probably intended for measuring speed and trajectory of guided missilesat the firing point; (iv) various small and delicate parts of guided missiles, fastened to cardboard by fine wires, with full description of each item attached to the cardboard.
c. The boxes were buried near VERONA by the FELDWEBEL, [censored] and a third soldier.(The site has been explored. Parts of wooden boxes or of a wooden box, fragments of paper and anampullae (phial) marked U234 or U235 were seen by Captain SNEIDER and [censored]. The rest had vanished). [...]BREE may be the same "person with a French name" (BOREU?) who worked spasmodically on "electric fuses for guided missiles" at TUCHELER HEIDE in 1943 and 1944. Thus he would know valuable details on the combined "guided missiles–atomic energy" research and development and would perhaps know where the missing documents were sent, whether they went first from TUCHELER HEIDE to BERLIN, as [censored] vaguely stated, and thence to SD, POTSDAM,and to Italy.
24. Ingenieur KRUEGER should be traced and brought at once to ECIC. He may give us valuable information on the combined "guided missiles–atomic energy" program at TUCHELER HEIDE in1943 and 1944 and may know exactly where documents and instruments have been sent. [Censored]obtained most of his information on activities at TUCHELER HEIDE from KRUEGER, in 1944.
25. Prof. Dr. NIELS [Walter Nielsch?], now said to be in the United States, was, according to [censored,] concerned with chemical and atomic problems at TUCHELER HEIDE and produced a number of atomic bombs, weighing from 1 to 5 kilograms. NIELS should be traced and questioned in detail.
26. Prof. Dr. HUETTEN. Present whereabouts unknown to [censored.] He should be located and brought to ALASKA for questioning. According to [censored] he was the originator of the combined project of research and development of atomic energy and guided missiles at TUCHELER HEIDE.This project was named "Aktion HUETTEN" after him. He was transferred elsewhere, probably in 1943 (see paragraph 7). (Wait, what? Not Heisenberg? Who was Huetten? The Tucheler Heide region is in Poland and there are additional documents that describe German work on atomic weapons being done in and around the area. -- WP)
27. Prof. Dr. HOFMANN, successor of HUETTEN as chief of the combined program at TUCHELERHEIDE, is now at "ALEXANDROWKA Kononien" near BAKU where he is continuing his formerwork. [...]
28. Prof W. Von BRAUN should be re-interrogated on the following: [...]
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The known prototypes appear to have consisted of at least three designs: 1) a fusion-boosted fission, implosion type strategic nuclear weapon with a maximum blast yield of 100 kilotons. Two of these were tested near Ohrdruf, Thuringia, Germany in early March 1945 but with minimal loads of HEU--just enough to achieve supercriticality in order to test the troublesome implosion bomb fuzing that had given German scientists many problems. 2) A more standard (by Allied measurements) basic fission bomb with a yield of around 40 kilotons. 3) A weapon of uncertain configuration that may or may not have been extremely heavy. There is conflicting information about this design, specifically by way of the Italian journalist Luigi Romersa. More on that later if you want additional detail.
Beyond these were the three nuclear weapons produced in an underground complex just inland and uphill from the Rhine River in an area known as the Luneberger Heide. The most likely configuration for these was that they were tactical or battlefield nukes with a yield of between 130 tons and 3 kilotons, per Himmler's adjutant, Werner Grothmann.
It is known that at least one (1) incomplete strategic nuclear weapon was captured by Patton's Third Army when that formation overran Ohrdruf in April, 1945. Grothmann described demands for an immediate nuclear strike against the United States by "hotheads from the political side" in the wake of 1) the successful detonations near Ohrdruf and 2) the test flight of a prototype ICBM that had sufficient range to strike the American East Coast. All of these occurred in early to mid-March, 1945. This would-be longshot would have been attempted with another prototype missile and a "hastily assembled warhead".
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Regarding why the (probably---) small number of German nuclear devices were not used in combat, Grothmann gave the following explanation:
Heinrich Himmler's chief adjutant Werner Grothmann on why nuclear weapons were not used in combat [Krotzky 2002]. For a discussion of the background and reliabilityof this source, see p. 3120. See also Grothmann's testimony on pp. 3120, 3642, and 3757.
"[p. 13] Himmler had in any case come up with a report, and a consultation took place regarding what we could make now. The one problem was the small quantity [of fuel] and always still the uncertainty of how it would work in action.The other was the question of the real political effect.
[p. 13] Some said that a direct hit on Moscow must be the first goal. But this has been countered by the argument that this would not change anything on the eastern front.[p. 14] If we were now to use such a weapon on Hitler's order, for example to employ it on London, a completely new situation would arise, but not in our favor. If the weapon's im[1]pact corresponds to the calculations, important parts of the political and military leadership will fall, but many other levels that have been relocated outside will be preserved. There are heavy casualties among the civilian population,and when the horror has subsided, it is clear that the supply of potential British troops in the Reich is still possible via their ports and is still under their control. Besides, the British are also on our territory. And the most important argument: with us, no one really believed that they would then withdraw. Quite the contrary! We could picture their reactions to our population. The other side, which must also be considered, is the Americans.
[p. 15] At the meetings I attended, or about which I learned in hints, no one was so crazy to use a weapon which could no longer help us, but would only make things even much worse.
[p. 16] So, the first point was that the decisionmakers had to know how they personally fared if a completely new, terrible mass-destruction weapon were deployed by our side and achieved its effect, but the war were nevertheless lost by us. What the victors would then read out of the Geneva Convention was clear. The second point was that: At that time, the demand for unconditional surrender had long been on the table. And that was the result of the normal war situation. What would have happened after the use of our atomic bomb? You are certainly familiar with the ideas of Morgenthau. Everything would be much worse."