Thanks for your input all.
I have found another link which perhaps might be of interest to some of you here regarding China and various aspects of it's naval history:
https://rwhiston.wordpress.com/2010/04/15/5/
More or less in line and with more detail with what I have written in this thread.
Assuming China somehow avoids getting involved in WW1 on the wrong side, it must also somehow avoid getting into a war with Japan (so it can continue to develop and enlarge this navy). I have read both here and other places regarding the "21 demands" of 1915 which apparently irked the UK and US, so if China plays it's card right in this ATL, they might jump to declare themselves on the side of the Entente, maybe even send some token troops and ships to Europe, what will the japanese do then if China flatly refuse the demands, go to war against China and risk themselves in a war with the Entente and probably the americans? I can see the two sides trying to get into the favours of the Entente, if China sends troops and ships then Japan will likely increase it's own commitments too.
This brings up a problem Yüan Shih-k'ai is a bottleneck. He has to go. I don't know who can replace him. I do know that Chiang kai Chek knows as much about sea-power and how to attain and apply it as the mythical man in the moon. The best answer I mean here is that China cannot look beyond herself to attain any form of sea-power. She has to attain for herself the basic tools and then build from her native base the fleet you want her to have in the OP. That is, the problem defined, going to be difficult. Every other power has at least a 100 year lead in the learning curve. The Japanese seem to have truncated the process, 1868-1908 (50 years), but realistically they imported the industrial revolution and the social system about the same time China went shopping for the polyglot fleet that she did not know how to use. China did not embrace the social change that she needed to figure out how to build European style institutions that go with European style technology. And she did not create a technology of her own to compete with the foreign devils.
Politically, i can see this increasingly powerful China standing up to Japan as being seen useful by the western empires (UK, US, France), as long as they are busy with each other it removes the threat to their own interests in the region, so they will likely do their best to manipulate the rivalry between China and Japan, but probably aiming not to give one or the other side the edge.
Politically, the current modern China, which is trying in the 21st century to pull a Tokugawa reform movement on its own (Zhou en Lai around 1974) in reaction to the Maoist revanchist Doaist style Great Leap Forward, is as I've earlier noted, is about where the Japanese were in WW I. This is not a good thing. Two choices as you point out, lie before China and it is a coin flip to see which way it will go.
We can attempt a PoD around 1909 and suppose Sun Yat Sen will absorb more of his Hawaiian experience than he did. He can be faulted for trusting Yüan Shih-k'ai and thus continuing and accelerating the very Manchu (Qing) corruptions that make 1920s China such an interesting criminal sociology study. If Sun Yat Sen can be convinced either by a trusted foreign advisor or by his own inner circle to do a Lenin, it will be hard on the ruling elite, and it will take on some of the attributes of the Russian Revolution, but the end result is a clean slate start that China can make around 1922 or ten years after the first attempt to govern. That is about how long a civil war should take before the KMT has the core areas of China under control.
This will be anathema to Japan and to many western nations. But we are addressing what is in China's best interests. As with many socialist revolutions, the top down control will be imposed by totalitarian means and by totalitarian ideologues. The tools, the Kuomintang and the army will tend to be the apparatii that will first benefit from the social and technological dislocations. Notice that China will not be a participant in WW I, except insofar as the Entente operates to neutralize German operations in China proper.
Any military issues the Chinese government addresses in its first plan will be to establish sovereignty where it can. Following the traditional Chinese model, this will be harassment, internal blockade and eventual waiting out the Xi Yang Gui Zi or (Yankees to befuddled Americans.) until they leave. (Hong Kong is a modern example.).
The KMT government (1928) can now theoretically go shop for its industrial base. They should look for things a government intent on asserting its sovereignty needs to have as tools. The shopping list is surprisingly not going to be naval per se, but more land oriented. The central government needs foundries, steel mills and at least a small precision tool and milling machine base. Guns from small arms to artillery first. Then aircraft and then automotive technology. In that order. Along with this base, China must send its best and brightest to friendly western societies to learn how the West does things. These Chinese must return and become the new technoratii, who will design and build the first Chinese versions of the formidable western tools.
In the interim, the Chinese, hire foreign experts to teach how to run, but not own the new Chinese tech base. Projected completion of these measures? 1938. Ambitious. From 1938, possibly 1935, the KMT can think about building a navy.
Anyway, to more technical matters, apart from the ships they could possibly have gotten before WW1 listed in the previous link, after the war i was looking at what possibilities there were to obtain truly modern ships, and there are lots: from Germany, any of the unfinished Mackensens, Sachsen etc., from Italy the Caracciolos, from France the Normadies, from UK some of the Hoods(!), and/or some of the follies, maybe some from the US too, not to mention many other options for second hand capital ships, as well as cruisers, destroyers, subs and other craft. Of course, money could be a problem, together with the willingness (or being allowed to in case of Germany) of some of these countries to sell to China, but then in the post-war period, any amount of chinese money would be a godsend for any of these countries.
One market only. Hired experts and it will be the United States. Not because the United States is a friend, or any less imperialist than any other nation infringing on Chinese sovereignty, but because the Americans are Japan's enemy, and "enemy of my enemy" is so Chinese.
Also, if China would go this way, surely sooner or later they would have modern shipyards built to be able to progressively built ships of all classes in the country. Could that China possibly become a naval powerhouse like it is today? Not impossible i would say.
Can China become a naval power capable of self defense? 1940 maybe. If the KMT pays attention to first things first. Build the army and the air forces to protect the core sovereignty, then look at the seas.