I can only highlight/expand on what ceccherini stated above.
The RM did develop a design for something past the
Littorio-class, but was not exactly highly enthused about building more and larger battleships. In particular Admiral Cavagnari, who was the navy's Chief of Staff and Undersecretary of the Navy at the time, was very conservative on this matter and did not want to do anything other than order more of an already proven design that could be built rapidly - which was a reasonable stance, considering that
Impero and
Roma were fairly rapid builds (if not for the fact that
Impero was moved from Genoa, she would have been completed in August 1941, and would have become operational before the end of the year).
As far as what design was actually being worked on - the navy had actually been working on a design since 1935 that could have 'succeeded' the
Littorio-class, should the navy decide to move in that direction, and was developed in the context of the infamous 'Breakout Fleet' program of 1935/36. This was intended to be a 41,000-ton standard displacement battleship armed with 3x3 406mm guns, 4x3 152/55, four aircraft with a covered hangar, and a top speed of 32 knots - which very rapidly grew to a 42,000-ton ship.
This design, as it was in 1935, is what ended up being used by Ansaldo's project office as the basis for the design of the 'UP.41', a derivative design developed for the Soviet Union through to mid-1936. In the meantime, the RM continued to develop the design from 1935 up to the summer 1941, when all work on the design ended for good. The design had grown over time, much as was the case with the
Littorio-class (and even went through some of the same evolutions, such as the replacement of the planned 100/47 AA battery with the 90/50 in 1936), and so ended up at a displacement of 45,000 tons by 1939.
The degree to which the design was ever seriously considered for
procurement is limited. Not many secondary sources bother to report the exact composition of the 'B' programme of the 1935/36 'Breakout Fleet', which was the most radical version, calling for nine battleships and three aircraft carriers. However, Enrico Cernuschi & Vincent O'Hara in their article on the subject in
Warship 2006 ('The Breakout Fleet: The Oceanic Programmes of the Regia Marina, 1934-1940) report that this version of the program hoped to procure two of these ships, alongside a trio of smaller 17,500-ton 'pocket battleships'. These would be added to the existing force then building of the two
Littorio-class battleships and the two
Conte di Cavour-class battleships then under reconstruction. The conventional battleships would operate in the Mediterranean, while the 17,500-ton ships would be based in East Africa as an independent raiding force (along with supporting ships). All ships should have been procured by the end of 1942.
This plan was presented to Mussolini, but went nowhere due to the great expense. The 'backup' program ('Programme A'), which also failed to gain approval, only called for six battleships (and no carriers) - according to Augusto de Toro, this plan saw only six battleships being reached by the order of a single additional
Littorio, and the reconstruction of
Duilio or
Andrea Doria, in order to create two homogenous divisions of three battleships. The end-date for this program was still 1942.
The next time the question of battleship procurement came up, it was 1937, and it does not seem that the larger 406mm battleship was raised for consideration at all. Early in the year another long-term plan was drawn up, intending to outline naval procurement from FY1937/38 to FY1943/44. This called for a force of eight battleships and 'three' carriers (actually only two 15,000-ton carriers, in addition to the 4,880-ton seaplane carrier
Giuseppe Miraglia). At no point that I am aware was the '41,000-ton' battleship raised as an option - the planning assumptions were always that an extra pair of
Littorio-class battleships would be procured, and the reconstruction of the
Duilio-class battleships initiated. The light cruiser force should have also reached 20 ships totaling 120,600 tons (standard displacement) via the construction of six additional
Duca degli Abruzzi-class light cruisers (plus eight scout ships of 2,500 to 4,000 tons, but I'm not sure if those were counted under light cruisers or with the scouts). Such as it was, this program was not executed, though battleship construction policy did effectively follow this program - at the end of the year,
Impero and
Roma were ordered, as was the reconstruction of
Duilio and
Andrea Doria. Additionally, the twelve oceanic scouts/light cruisers of the Capitani Romani and sixteen oceanic submarines were ordered, all coming under the aegis of the FY1937/38 program and 1938 supplemental program (which sort of ate any prospect of a separate FY1938/39 program).
The last time the '41,000-ton' battleship seemed to get a chance at being ordered was in 1939. Early that year was the last gasp of any major fleet plans, namely one that sought to outline construction from FY1939/40 to FY1944/45. This hoped to procure a pair of battleships of 45,000 tons, two 15,000 ton carriers, twenty 2,100-ton destroyers, twenty 1,000-ton torpedo boats, and some 60 submarines. Admiral Somigli was apparently in favor of it, while Cavagnari preferred the prospect of ordering a third pair of
Littorio's instead, but in any case the concept of ordering two new battleships quickly fell apart as tensions rose in Europe and the reality of Italy's shortfalls of raw material imports bit. In any case, the navy seemed to place a higher priority on procuring a pair of 15,000-ton carriers than on new battleships - the navy's planning office had in fact been requesting the two ships each year since the 1935/36 programs. These were ultimately pushed back to the FY1940/41 program (which obviously never materialized), and it was hoped that the ships could be laid down in the following year or two and commission in 1943 and 1944.
Such as it was, the actual FY1939/40 program, approved in April 1939, was split into two parts.
- To reinforce the fleet; twelve '1,500-ton' destroyers (enhanced Soldati with a dual-purpose armament), twelve 635t medium submarines, and six 1,100t minelayers.
- To form a raiding force that would operate out of Italian East Africa and attack British and French traffic in the Indian Ocean; three light cruisers of the '8,000-ton' Costanzo Ciano-class, two 8,500-ton fast oilers, twelve 1,350t oceanic submarines, and six destroyer escorts to protect the planned base at Chisimaio (evolved Orsa/Pegaso types, with diesel propulsion).
Such as it was, the FY1939/40 program was never executed, though nor was it officially canceled - some of the flotilla ships and submarines ended up being ordered later as part of wartime programs intended increase the navy's escort capacity, or to replace losses of destroyers, escorts, and submarines.
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Stepping beyond the bounds of what
actually happened, and towards summary and speculation - in broad strokes, though the potential construction of a 45,000-ton ship was not exactly ruled out, the RM as a whole was not really in favor of it, so long as existing battleships were adequate to their needs - flatly speaking, they had areas that needed reinforcing more than the main battleline. The navy's planning office was adamant about the need for carriers (even if Cavagnari was far less enthusiastic about them), and likewise also desperately wanted a larger force of modern destroyers (by which they meant ships of the
Maestrale-class and later) suitable for escorting the additional battleships that had been procured after the modern cruiser program. If not for the crisis in supplies of raw materials and the tension in Europe in 1939, I strongly suspect that the navy would have ordered carriers in FY1939/40 or FY1940/41 and tried to complete both of them before 1944. By necessity these would have had to have taken up two of the three slips available for capital ship construction in Italy. Of course, neither of these conditions were filled, hence why it did not happen in reality.
That said, the wildcard in any scenario favorable to further Italian capital ship construction is future European construction. The Regia Marina was hoping that the upcoming disarmament conference in 1940 would result in a limit of 40,000 tons for battleships. So long as the rest of Europe remained by that limit, Italy's position of maintaining their battleship force of 'only'
Littorio's - '35,000' ships that were 40,500 tons standard in reality - would have been perfectly valid. After all, at the time the French were only ordering additional '35,000-ton' ships of their own (
Clemenceau &
Gascogne), and the new British battleships of the 1938 programme were 40,000-ton ships (
Lion &
Temeraire).
This vision was very flawed, however. They made this assumption based on their discussions with the Germans about coordinating future capital ship constructions, and the Germans were lying through their teeth the entire time. They made the Italians aware of the planned H-39 battleships in early 1939, but informed them that they were for one or two 40,000-ton battleships. The actual scale of the program - Plan Z - called for six 52,600-ton battleships.
France, who had long been Italy's main concern for naval construction (and vice versa), had largely drawn the same conclusions as Italy and were in fact suffering under many of the same industrial and infrastructure limitations. They also did not want to go above 40,000 tons, and saw 45,000 tons as something that only the Pacific navies should/would concern themselves with. Of course, like the RM, the Marine Nationale was still prepared to explore a 45,000-ton design, which was the maximum they could feasibly build (again, like Italy).
That said, in mid-1939 the French learned of German plans for the H-39-class - though their intelligence services had concluded they were 40,000-ton ships - and consequently Darlan did set the navy to study battleships of 40,000 tons displacement or more, and with guns of 400 to 420mm. This spawned a new design series of battleships - those we generally call the 'Alsace'-class (no name was officially assigned, though 'Alsace' was the first of those raised as possibilities) ranging from 40,000 to 45,000 tons. Ultimately, they opted to authorize two 40,000-ton battleships with 3x3 380mm guns in April 1940. Neither ship was laid down, of course, due to the Fall of France in May-June 1940, but the first was intended to be laid down in 1941 when the carrier
Joffre was launched (actually replacing the planned second carrier,
Painlevé), and the second before the end of 1942 (it was dependent on the new building dock at Brest being completed on time).
This leaves us with a rather interesting question. Should Europe not have exploded into war in 1939, but rather several years down the line (say, 1942) - Italy would have been faced with France building an extra pair of modern 380mm battleships, giving France a 3:2 superiority in this type over Italy should no new Italian battleships be built. Likewise, the scale of German (the 52,600-ton H-39's) and Soviet (59,150-ton
Sovetsky Soyuz-class) construction would have become apparent. If nothing else, I think this is likely to spark a more serious look at building the 45,000-ton battleships than they gave in reality, because the cat would well and truly be out of the bag on any kind of 40,000-ton limit at that point.