This was being discussed over a couple of naval boards, sa I recall. From what I could see, general consensus was that in the purely naval sphere, developments may not be all that revolutionary.
By the time of the treaty, the USN had already decided that it needed CVs approximately equal to the Lexingtons in capabilities. It seems that US Congress would be unlikely to fully fund the 1916 program in full. The South Dakota class would probably have at least a couple of units completed, but the Lexington CCs are less likely - it's probably that you'd get one or two competed as CC and one or two as CVs, with a total of three to four.
The IJN would have a crack at the 8-8 program, but the Great Kanto earthquake would still utterly root their economy for years to come.
The British are probably the most interesting, in that IIRC they had enough economic capacity to complete the G3-class battlecruisers.
In the political sphere, it becomes much more in Britain's interest to renew the Anglo-Japanese alliance to strengthen their position in the far east. This would of course continue to state that the UK would not be drawn in to a Japanese war of aggression against the United States. However, it does make it much more likely that the UK and Japan cooperate more closely in the far east, particularly in keeping order in China and suppressing piracy. You could see a joint Anglo-Japanese move into Manchuria aimed at establishing a joint protectorate, which would very much be in Japans interested economically after Kanto, but which would prevent them from becoming diplomatically isolated. A strong relationship with Britain would also help to keep the ultranationalists and lunatic fringe of the Army and Navy under control.
In Europe, Italy and France would still be too economically poorly placed to realistically commit to building programs much in excess of what they did OTL. Italy may choose to build new ships rather than refitting their old dreadnoughts, but the Caracciolos had already been cancelled and the existing work scrapped by the time the treaty was signed. France was exploring the quad turret even before WW1, so they may look into something along the lines of the Gascogne or even the Alsace classes. The Dunkerques were explicitly a response to the German BCs, and so any analagous ships would probably be strongly contingent on German shipbuilding.
You would probably see, over the course of the 20s, the scrapping by the major navies of all ships with a main armament smaller than 13.5", followed by a scrapping of ships with armament smaller than 15" in the end of the 20s and the 30s. Many of these ships would be put up for sale before being scrapped, which could see second-tier nations like Brazil, Chile, Argentina, Spain and Greece snapping up some less-capable ships at bargain prices.
Psychologically, the impact on Japan of not being explicitly declared a second-rate power would be immense, and having limited expansionism sanctioned would bleed off quite a few of the more aggressive leaders. I wonder how it may affect the Japanese involvement in the Russian Far East - a more confident Japan and a strongly anticommunist Britain with an interest in the Far East might end up melding the Manchurian protectorate and the White Russian forces together.
Britain would probably feel much more secure with a full-strength fleet at its disposal and no real competition - German has been neutered, the Russian/Soviet Navy is no real threat, and Japan, France, Italy and the United States were most recently allies. The G3s are a remarkably powerful design, and a second set in the late 20s, possibly with a more conventional turret layout, would continue to keep British naval power well above the rest.
Germany would still be bound by the Versailles limitations of a maximum of 10k ton ships, which historically was fudged into the 16 000+ ton Deutschlands. Without the WNT, and its 35 000 ton standard for battleships, the Scharnhorsts aren't really that viable as fleet units. An Anglo-German Naval Agreement is also less likely, as the UK would be less focused on putting everyone on a equal footing, and more inclined to see the newer German fleet units as aimed directly at the RN and commerce warfare.
Carrier warfare would continue to be developed - the RN, IJN and USN all saw the potential well before the end of WW1, and there may be fewer ruffled feathers if the Admirals are not forced to give up battlewagons to get CVs. The USN in particular was looking towards Carriers to be the scouts of the fleet, and their designs from before the WNT were based on the ships being able to carry strike bombers.
Another interesting area would be cruisers - Britain had identified a strong need for a large number of trade protection cruisers, and without the WNT Heavy/Light cruiser division there would not be the tendency to build to the limit as was seen in OTL - nations wanting line of battle ships are free to build them.
My summary, then, would be that you'd probably see a strong partnership between Japan and Great Britain - though Britain would probably still see Japan as being somewhat inferior and more of a regional power than itself - with joint commercial and military missions in China and Manchuria. The USN would be strengthened by some South Dakotas, some Lexington CCs and some fleet CVs, but weakened by the presence of large numbers of Standard ships with their limitations, and a very top-heavy fleet, combined with a Congress unlikely to authorise further large building programs. Japan would be more confident in its place, and would probably have a more stable and strong parliamentary democracy. The UK would have a stronger navy, and the British would probably draw confidence from that strengthening of their place in the world. It's probable that the Dominions would have strengthened navies to boot.
Things could start to get a little interesting if the far-right rise to power in Europe as in OTL, or if the USSR decides it needs to compete in the world. Ultimately, the only major changes would be that Japan is less likely to fall to the far right, the UK is likely to be more confident, and the US less likely to engage in new construction - anything other than that is just altering the equipment rather than the outcome.