Now, assuming the Tillmans came to be, what would they be named? And more to the point, what role would they have? Here's what I'd figure.
Any operational Tillmans would require the Washington Naval Treaty to not happen at all, as it would make a 80,000-ton giant impossible. This has the after affect of not stopping a growing arms race between the US, Britain and Japan.
Now, assuming these are authorized in 1919-1920 and begin construction, with hull numbers BB-49 for the first. No Washington Naval Treaty means that the US' battle fleet probably includes the South Dakota class (the first one) and the Lexington-class battlecruisers, which the Tillmans would more likely work with - their 25 knot+ speed allows them to be the heavy backup to whatever the Lexingtons can't handle, which admittedly isn't much. Keeping the names from IOTL's South Dakotas and remembering that the name of the South Dakota class was already chosen, the likely BB-49 (the first Tillman) would be Indiana, followed by Montana and North Carolina. I can't see any more than three being built in any scenario.
The first Tillmans are finished in 1923-24 and commissioned, with Indiana, Montana and North Carolina being by some margin the largest battleships in the world at the time. The six Lexington-class battlecruisers are also built, and the Tillmans are assigned to many of the same duties as the Lexingtons as the Tillmans are considerably faster than the Standard design battleships. The Tillmans' casemate mounts prove to be difficult to operate and useless in heavy seas, and they are removed in 1920s refits.
The Tillmans force other nations to move ahead with bigger and heavier designs. Japan finishes its eight-six fleet in the early 1930s, with the Kii, Amagi and Nagato class vessels all seeing completion. Britain winds up scrapping some of its oldest vessels and calling on Commonwealth help to build the Royal Navy to the point where it can keep up with the Americans - but even then, the USN has a decisive advantage over everybody else.
The Tillmans are refitted and modernized in the 1930s, substantially increasing their AA armament, among other improvements. The Tillmans, however, suffer from a relative lack of armoring, and this cannot be improved without the beasts being unable to pass through the Panama Canal. Despite this, the Tillmans are an active part of the USN right through the 1920s and 1930s.
Knowledge of Germany's Bismarck keeps Indiana and Montana in the Atlantic, while North Carolina goes to the Pacific. North Carolina is severely damaged at Pearl Harbor, taking particular damage from aircraft bombs - but she, like all the other vessels at Pearl Harbor save Arizona and Maryland, is repaired and sent back into the Ocean. The speed of the Tillmans and the Lexington class battlecruisers makes them useful parts of the fleet, even as the first new modern designs, the Iowa (OTL North Carolina), Missouri (OTL South Dakotas) and Ohio (OTL Iowa) start to arrive in the fleet.
Japan's immense Yamato is a primary concern for everybody, but the Japanese, knowing of the Tillmans' massive battery of 15 18" guns, is tasked to find and destroy the Tillmans. Following Pearl Harbor and the loss of many of the American fleet for a time, Indiana and three Lexingtons go to the Pacific. That results in Indiana being called in to back up two of her smaller sisters during the famed battle with Kirishima. The Japanese ship damages the two smaller battleships, but Indiana's giant 18" guns blast the Japanese ship to bits.
North Carolina, substantially rebuilt, returns to the USN in mid 1943, and having had much of its superstructure rebuilt and many improvements made, she is fearsome - and the Japanese know it. The same major upgrades soon are fitted to Indiana and Montana. They serve out the war with distinction, though Indiana is the only one to fight an enemy battleship, though North Carolina was dispatched to go after Yamato if the bombing attempt on her as she move to reinforce Okinawa failed. In many ways, North Carolina's crew was a little disappointed they never had the chance to go toe-to-toe with the Japanese beast. The war over, all three are decommissioned in 1946 and remain in the reserve fleet, and all are eventually scrapped in the early 1960s.
