One big issue is that wood is not practical for ships of this nature. The largest wooden ship ever built was the schooner Wyoming from 1909. That ship could carry about 6000 tons of cargo, but the wood construction also meant that it "tended to flex in heavy seas, which would cause the long planks to twist and buckle, thereby allowing sea water to intrude into the hold". The had to run pumps to keep out the sea water. In comparison, Liberty ships, which were relatively cheap and quick to make, could hold about 10,000 tons or more of cargo and were relatively seaworthy.

Basically, the minimal savings in steel (which the US actually had quite a bit of) would not have been worth the drawbacks inherent in wood construction.

Beyond the issues with large wooden ships highlighted by /u/radomirputnik, there are other, logistical issues with building ships out of wood, especially en masse. Firstly, compared to metal, wood is difficult to work and join. Bending a sheet of steel to shape can be done by unskilled labour, while welding can be taught quickly and effectively. Woodworking skills, by comparison, take years to gain. Shaping a plank is a difficult business, while knowing how to nail them to make a hull is tricky. Because the vast majority of shipyards, and all of those making large warships and cargo ships, were working in steel, there was a small labour pool capable of working in wood. It would be difficult to expand this, and all but impossible to do it quickly. Meanwhile, there was a large pool of workers capable of making a ship out of steel, and it could be expanded quickly. Secondly, wood is not really more available than steel. Building a ship out of wood requires a lot of wood. Much of this wood needs to be in particular shapes - to make the strongest frame for a ship, you need pieces of wood where the grain follows the curve of the components you need. This requires trees that have grown in certain ways, that are comparably rare. Additionally, not every kind of tree is useful for shipbuilding. For the frame, you need hardwoods like oak or teak, which are similarly rare, or in the case of teak, need to be shipped from India. That said, small craft were frequently made from wood. Such craft were constructed mainly by yards that had been building yachts and pleasure craft for the civilian market in peacetime. Many of their workers had experience with wood working from this, and it made sense to capitalise on this.

Slightly related question: I'm vaguely familiar with attempts to build merchant ships out of cement (at least in the First World War). Since it ultimately didn't catch on, what were the drawbacks with this?

Thanks for the knowledge