So, this thread has sent be down a bit of a rabbit hole in terms of pre-WNT preliminary battleship designs. Here's what I've found so far, and my thoughts on what priorities future Royal Navy capital ships would have in this situation. The focus here is on what the US and Japanese navies were building/would have build, as these are the obvious rivals to the royal navy in the 1920s. Anyone else please chime in with extra info or opinions!
Pre-WNT Ships in construction or design:
UK:
4x G3 class battlecruisers
4x N3 class battleships
No preliminary successor designs known. I don't think any serious design studies were started pre-WNT, but I would be happy to be corrected on this!
US:
6x South Dakota Class (1920) battleships
6x Lexington Class battlecruisers
Norman Friedman's "US Battleships" states that the 1922 battleship design was intended to be similar to, but larger than the South Dakota class. As the armament was kept the same (although an 18" gun was being developed), I expect that the extra displacement would have been used to improve the armor over the South Dakotas, as their horizontal protection was especially on the thin side.
There were also a number of US fast battleship designs produced, however these were rejected as it was thought that they would make the rest of the battleship force obsolete (despite the fact that the fast battleships being built by Britain and Japan may have already made this the case). These would have been armed with 12x16" guns the same as the South Dakotas, and made 30 kts, the armor scheme would have been reduced and the hull lengthened.
Japan:
For Japan we probably have the clearest idea of what the future construction would have involved due to the 8-8 fleet plans:
2x Tosa class battleships
4x Amagi class battlecruisers
4x Kii class battleships
4x "No. 13" class battleships
The focus here is entirely on fast battleships, with the Amagi, Kii and No. 13 class all making 30 kts or near enough. Armament is 10x16" aside from the No. 13 class which is thought to have had 8x18" guns, although it is uncertain whether this was the final design, or whether another of the alternatives with 12 (or more) 16" guns would have been used. All were armored on the light side against 16" guns
Overall, the picture we see is that a potential future opponent will either be an enlarged US slow battleships design, likely with 12x16" (or possibly 8x18") and heavy armor against 16" guns (similar to that found on the N3 class) making 23 kts. As the N3 class are already more than a match for the South Dakotas, I foresee only an improved N3 design required to match the larger slow US battleship designs
On the other hand, we have the fast battleship designs, which for both the Japan and the US follow similar lines: 30 kts, 10-12x16" or 8x18" guns, and lightly armored against 16" gunfire. The G3 has a slight advantage in terms of both speed and armour, but is at a disadvantage in terms of firepower. I think that most of the effort would be directed here at developing a fast battleship from the G3 design with improved armament, but maintaining a well balanced protection scheme
(On an in game note, I see that in your save the German Mecklenburg class BBs have very similar characteristics to the South Dakota class, and Japan is also building fast battleships with 10x16" guns...)
Potential operational and design requirement post-G3/N3 classes
This section discusses what - in my opinion - would be the design considerations for a successor to the G3/N3 class, considering both the design process that went in to the G3, N3 and Nelson classes, and the lessons learnt from those ships in service. As this is very much subjective, I welcome others perspective on this, although here I've tried to stick to frame of mind of the Royal Navy in the early 1920s.
Operational requirements:
There are two main scenarios for a 1920s war that the Royal Navy has to consider; firstly, a war with the USA (potentially with Britain and Japan as allies), and secondly a war against Japan. Both of which would involve the Pacific as a significant theatre of operations. As the UK is on the other side of the planet to the Pacific, this adds significant requirements (which were also involved in real life in the design of UK WNT aircraft carriers, particularly Ark Royal). The limitations on beam and draft required to use both the Suez and Panama canals remain from the preceding G3/N3 classes. Range isn't too much of an issue, as the British empire still has a large number of overseas bases for refueling. Displacement is an issue however, as enlarging dock facilities across the the world would be both an extremely expensive and difficult task. As the G3/N3 class already pushed up against the maximum displacement limits, I can only see an increase in displacement of a few thousand tons - if any at all - as being realistically allowed. In addition, as an operational loss for the Royal Navy fighting in the pacific would be much more costly than for the USN/IJN due to the greater difficulty of returning to home bases for repair, a greater emphasis on survivability - particularly including protecting the machinery spaces - is required.
Speed:
In my opinion there are only two options here: 23 kts or 30 kts. Although the preliminary designs for the N3 class called for a 25 kt speed - matching the Queen Elizabeth class - the advantage of this is no longer present in the 1920s. Whereas the 25 kt speed of the QE class enabled it both to have a tactical advantage over a 21 kt battle line, and also catch the slower German battlecruisers, the increase in speed of the US battle line to 23 kts, and the IJN battlecruisers and fast battleships to 30 kts means that neither of these advantages remain. As a result, any increase in speed over 23 kts that does not match the 30 kts of the fast battleships offers little advantage at the cost of a lot of displacement. It was for this reason that the N3 class (and Nelson class) were dropped to 23 kts, and I see little reason for this to change. The G3 class were given a 31-32 kt speed in order to catch the faster Lexington class battlecruisers, but as I expect that the US would not have ordered another class similar to the Lexingtons, the requirement for greater than 30 kts would not be needed for a successor to the G3s.
Armour:
Both the G3 and N3 class were heavily armoured compared to their contemporaries, with 14-15" of inclined belt armour and 7-8" deck. Although the latter in particular is slightly excessive for the 1920s, I cannot see the Royal Navy accepting a reduction in armour post Jutland. I cannot see much change occurring from the armour scheme of the G3/N3s (and the underwater protection as well for that matter).
Armament:
This is, I believe, where the most major changes would have occurred. The armament of the G3/N3 classes was designed with the principles of a lightweight, high velocity shell based on flawed test results. I believe the poor performance of these guns would have been noticed once the G3s/N3s were in service, as they were with the 16" guns on the Nelson class, and a return to the heavy shell principle desired. I believe the 18"/45 gun would have been salvageable as it would still offer an improvement over the 15"/42, and because a heavy shell had already been developed for the 18"/40 gun used on Furious. The 16"/45 as used on Nelson however was very disappointing, compared to the 15"/42. As a result, I think the designers would reconsider the same choice of guns initially discussed for the G3 class for use on a G3 successor:
15"/50
16.5"/45-50
or 18"/45
Out of these, I cannot see the return to a 15" calibre weapon. On the other hand, adopting the larger 18" gun would mean a reduction to 6 guns in two triple turrets (as with the H3 designs), and with the emphasis on good ladder shooting fire control in the RN post Jutland I can't see them accepting a capital ship with fewer than 8 guns. That leaves the 16.5" guns, which were originally planned for the G3 class, before being switched to 16" late in the design (if anyone has a source on why this switch was made I'd love to know!). Tests of this calibre could possibly be performed by boring out and re-lining the 16"/45 to 16.5".
In terms of mountings, I believe the triple turrets of the G3/N3 class would have been maintained, as without the aggressive weight saving required for the Nelson class these would have likely had fewer teething issues. Quad turrets are out of the question as they could not be used with guns greater than 15" without exceeding the limitations on beam. One possible change is that with the G3/N3 class in service, the drawbacks of the X turret location would have been noticed, and a more conventional aft location of the X turret desired in future ships (this conclusion seems common to both the British and French navies after building ships with all forward armament). One possible placement of the X turret may have been in between the boiler rooms and engine rooms, as this would have preserved the short shafts which made for substantial weight savings in the G3 class without having the funnels restrict the firing arc to aft (although I'm sure this would also have its own engineering issues).
Overall, in my opinion the changes to the N3 class would have been minor, with some changes to allow for heavier 18" shells and a possible relocation of the X turret to a more conventional aft location, with weight savings made elsewhere to account for the increase in displacement. A G3 successor would likely have more extensive changes, including a reduction to 30 kts, rearrangement of the turrets and machinery spaces, and a possible increase in calibre.
Whether an N3 successor is built at all would, I believe, be in response to the US 1922 battleship design. If the US built another, larger, slow battleship class then I believe 4 N3 successors would have been ordered in response. If the USN instead switched to fast battleships then the focus would be on G3 successors. In either case, an improved G3 is needed to counter the Japanese fast battleships, as the armament of the G3s is lacking in comparison.
With all that in mind, here's a possible in game successor to the G3 class. It's by no means ideal from an in game perspective, but it's how I believe a G3 successor fast battleship would look like in game. I had intended to limit the full load displacement to 60,000 t, but the higher required hp and high armour weights in game than in real life required both an increase in displacement and a reduction in armour thickness. It still enjoys a sizeable immunity zone against both 16" and 18" guns however. 17" guns are used to represent a possible 16.5" gun. Unit machinery would be nice, but this wouldn't have been feasible in real life with the proposed arrangement of the boiler and engine rooms anyway.