We already have CombinedFleet's Battleship Comparison:
http://www.combinedfleet.com/baddest.htm
I'll work from that and I'm putting it here so nobody thinks I stole anything.
In terms of raw firepower, Yamato's 460mm guns are clearly the leader, but they are certainly not the most efficient guns. Four 16" guns are in the race here: US 16"/45 Mark 6 (North Carolina, South Dakota), US 16"/50 Mark 7 (Iowa), RN 16"/45 Mark II (Lion), and KM 40.6cm/52 SK C/34 (H class). I think we can quite fairly judge these guns simply based on long-range deck penetration, which is governed by shell weight and muzzle velocity. The German gun had a very light AP shell and high muzzle velocity, giving 5.0" of penetration at 30,000 yards. The British gun had a light shell but lower muzzle velocity, giving 5.73" of penetration at 30,000 yards. The higher muzzle velocity of the German shell would also contribute to reduced accuracy. The American guns fired the same super-heavy AP shell, but the lower muzzle velocity of the shorter Mark 6 gave that gun better deck penetration than the Mark 7, 7.62" vs 6.65", due to higher striking angles (closer to vertical). The extra range of the Mark 7 guns likely compensated for the decreased penetration, although long-range radar-guided gunnery was extremely difficult at this time. Therefore, I'll call the competition for 16" gun a tie between the Mark 6 and the Mark 7. Four 15" guns are competing: MN 380mm/45 M1935 (Richelieu), KM 38cm/52 SK C/34 (Bismarck), RM 381mm/50 M1934 (Littorio), and RN 15"/42 Mark I (Vanguard). There was a concept for a 15"/45 for the KGV, but that never saw any real design. Right away, I can eliminate the German gun for having too high muzzle velocity and too little shell weight. I know the gun and the ship it went on was designed to a short-range, limited-visibility knife fight in the North Sea, but that's not the kind of thing you send a battleship out to do by the time WWII came around. The Italian gun has excellent specs, with great range and acceptable deck penetration (compared to the German gun), but its muzzle velocity was high enough to limit round counts below international norms and it was only accurate if the shells were made properly to exacting specifications, which doesn't happen in Italy. The French gun also has a problem in that the muzzle velocity it needs to attain long range negatively impacts deck penetration, but it does not suffer too much compared to the gold standard. The clear winner in the 15" gun range, even after two decades, is the British gun, with excellent penetration, reliability, and accuracy in return for range, although I think there exists a hard limit on practical 15" gun range at about 38,000 yards. There is only one 14" gun I'm going to consider, the RN 14"/45 Mark VII on the KGV. This gun had armor-piercing performance almost comparable to the German 38cm guns. The upgraded US 14"/50 Mark 7, which went on the New Mexicos and Tennessees and was going to go on the North Carolinas before the escalator clause, actually had superior armor-piercing performance than the British 14" gun.
Essentially, these firepower rankings limit our options for best ship to the US battleships, which are clearly ahead, the British battleships, which might be able to make up the ground they've lost, and the Richelieu, which I think was one of the most underrated battleships of WWII. The Iowas and South Dakotas carried essentially the same armor protection and were somewhat more heavily armored than the North Carolinas, so we can eliminate those two from contention. All three British battleships (KGV, Vanguard, and 1938 Lion) share the same armor profile, which focuses more heavily on citadel protection while accepting slightly less armor on the gun turrets and barbettes. I know the KGVs, however, had among the worst torpedo protection of all the Treaty battleships, which contributed to the loss of Prince of Wales to torpedo bombers. Vanguard and the Lions certainly had redesigned and upgraded torpedo protection, though I don't know how much additional protection could be provided on what was essentially the same hull. The Richelieus carried an armor scheme somewhat between the relative extremes of the American and British battleships, but apparently also carried the most efficient torpedo protection ever put to sea. A similar system on Dunkerque protected that ship from 1.5 tons of depth charges exploding alongside at Mers-el-Kebir. The Dunkerques and Richelieus carried very deep torpedo protection, rivaled only by the Yamatos and the Littorio's relatively ineffective systems.
This analysis leaves us with the Iowas, the South Dakotas, Vanguard, the Lions, and the Richelieus. Although the Iowas were excellent ships, they were far too large and expensive for the relatively minimal improvement they offered over the South Dakotas to be considered a good value. Vanguard and the Lions both suffer from the same problems with relatively obsolete main armaments. For Vanguard, the guns were excellent but the recycled twin mounts meant that the ship was probably fifty feet longer, and therefore significantly heavier, than she needed to be. The Lions' 16" guns fell quite far short of the Americans and their super-heavy shells. The design could certainly challenge the South Dakotas or the Richelieus in terms of efficiency if it was shrunk to carry 3 x 3 15"/42 guns with improved torpedo protection, but the British 16" guns were never world-beaters.
In the final comparison between South Dakota and Richelieu, I don't know if I can make an unbiased decision. The Richelieu's speed and torpedo protection would certainly make it the superior carrier escort, while the South Dakota's better guns and superior fire control would make it better in a battleship fight. Overall, in terms of efficiency and value, the prize would have to go to the South Dakotas.
