I have a few things to note that have not yet been covered in other answers.
- Flexibility: traditionally, wooden shipbuilding used a time-consuming process called Steam Bending to temporarily soften pieces of wood in order to plastically deform them into the necessary curved shapes. If a mage can do that faster with magic, construction could go substantially faster.
- Structure: A ship's frame bears most of the forces the ship is subjected to. In theory a hull could be made thick enough to withstand such forces, but to do so in our reality would require so much extra wood and add so much extra weight that it's never done. Instead, hulls are made as thin as is practical given the conditions they will face (coastal and river [brown-water and green-water] ships can have thinner hulls than high-seas [blue-water] ships, trading ships can have thinner hulls than warships, et cetera). The hull is mounted on a complex structural frame that attempts to maximize strength while minimizing weight. The complexity is partially caused by the fact that wood's strength against any force varies greatly by the force's orientation relative to the grain, and thus the frame is made of many short timbers arranged to be in the best direction for the stresses each will endure, and braced at the joints with more timbers that distribute the stresses across the entire frame rather than letting it concentrate at a single point. In your world, however, the ability to seamlessly fuse pieces of wood to the extent of joining their grain structure allows you to do away with a good portion of this structure and make your ships lighter and more spacious for the same strength. The densification helps with reclaiming a bit of interior space as well. You still need extensive bracing fore and aft and beam to beam to resist buckling, bending, and twisting, but not nearly as much. Care does still need to be taken with the directions of the fused wood grains, as mentioned. The hull would benefit by having layered grain structures orthogonal to each other for maximum pressure resistance - you could produce something like a Monocoque made of a fusion of Marine Plywood and Cross-Laminated Timber, which would let you have even stronger hulls for the same weight.
- Failure modes: the Strength of Materials involves more than hardness. You said that the magic system can improve wood's Impact Strength, but you also want to figure out what your magic can and cannot do in terms of Compressive Strength, Tensile Strength, and Fatigue Strength. Under normal conditions, wood cracks when any of these strengths are exceeded - and the more force it took to reach the breaking point, the more dramatic it will be when that force is released.
- Fire resistance: Densified wood tends to char rather than be consumed by fire, which will help limit the spread of shipboard fires. Ropes and sails will still be extremely vulnerable, unless your world includes some way of fireproofing them as well.
- Masts and Rigging: Mast failure by bending and ultimately cracking (known as a "sprung mast" if it doesn't just snap off completely) became a major problem as ships got bigger and got more sails, which put extreme strain on the masts. In our world, this was combated by adding extensive webs of "standing rigging", strong ropes that ran between the top of a mast and the frame of the upper deck in the same way that a modern radio tower is supported by guy wires. As masts got ever taller and the forces on masts became ever greater, even extensive standing rigging was often insufficient, and it became common and then standard for masts to be reinforced by wrapping them at intervals with tightly-coiled rope and/or iron hoops. These helped keep the wood fibers from separating under bending stress. With your magic, it sounds like it would be feasible to clad a vertically-oriented core with a layer of horizontally-oriented fibers joined into complete loops, which should offer comparable strength increases. There is also the potential of magically joining wooden beams horizontally between masts to add support in compression (compared to rope, which is only strong in tension) and diagonally from frame to mast to act as buttresses.
- Biofouling: All ships and boats (and any other structures in the water), no matter what they're made of, are subject to the buildup of a layer of aquatic life including seaweed, algae, barnacles, tube worms, and hundreds of other species. This rapid buildup makes the surface increasingly rough, causing turbulence and drag that can slow a ship by up to 10%. Ships need to regularly undergo maintenance to clean their hulls and counter these effects. Wooden ships also have to deal with Shipworms, a group of molluscs that bore into - and eventually destroy - any submerged wood. The only effective long-term protection against shipworms in our world has been to fasten copper sheathing to the hull up to the waterline (which also conveniently reduces biofouling, but inconveniently prohibits the use of iron nails as it hugely accelerates their corrosion). Other metals like lead and iron are notably heavier and corrode quickly, making them impractical. A sacrificial non-structural layer of wood was sometimes used, which temporarily shielded the structural hull from damage and needed to be replaced frequently. Various toxic coatings have also been used, but they pose a health hazard and need to be reapplied frequently to maintain their effectiveness. Anything magic can do to ease this problem will give any navy with magic a substantial advantage over any without.
- Cost: This magic has the potential to drastically reduce the cost of building and maintaining a ship, especially with regards to time. Doing away with nails will save a huge amount of money and time (as @Kepotx says, nails are very time-consuming to make without industrialization and/or metalbending, and sometimes a ship would need to have its hull completely disassembled just to replace corroded nails before the bottom of the ship falls off). A less complicated frame cuts down on construction time significantly. Being able to manipulate "living material" strongly implies the possibility of magically removing biofouling, and unless such magic is itself time-consuming and difficult/rare, it would almost certainly be easier than having to manually scrape off the fouling. If that defouling can be done from inside the ship, without needing to careen it or use a drydock, the time and effort savings would be enormous, as it could be done without taking the ship out of service at all. Repairing a ship, as @DarthDonut mentions, will be a much smoother process as cracks can be directly mended and holes can be patched without having to remove and replace damaged planks. Faster repair means greater reliability and less time out of service. Being able to turn small pieces of relatively low-quality wood into a single long, uniform, high-strength beam means that masts can be made from whatever trees are nearby, instead of needing to find trees with the desired height, thickness, and straightness, and again saving time and money.
Further Reading:
The Mechanical Properties of Wood (1914), by Samuel J. Record [Unrestricted webpage and e-book] - Details the strengths and weaknesses of wood under mechanical stress.
The Structures of English Wooden Ships (1993), by Trevor Kenchington [Unrestricted PDF] - A description of the structure and construction of English ships circa 1710, based on a contemporary book written by William Sutherland after a career in the Royal dockyards, particularly at Portsmouth and Deptford. Walks through the entire structure of a three-decked ship-of-the-line.
Illustrated Glossary of Ship and Boat Terms (1994), by J. Richard Steffy [Unrestricted webpage and PDF] - Provides definitions, descriptions, and illustrations of the nautical terms used in Kenchington's paper, among many others. Excerpt from Steffy's book Wooden Ship Building and the Interpretation of Shipwrecks. Published as part of The Oxford Handbook of Maritime Archaeology, most chapters of which are restricted to online subscribers and people who buy the book [Restricted e-book, hardcover, or softcover] but which also contains a great deal of additional information on ship design throughout history.
History of Masts, from the National Museum of the Royal New Zealand Navy [Unrestricted webpage] - Provides details of mast construction
The Elements and Practice of Rigging and Seamanship (1794), by David Steel [Unrestricted webpage] - Provides extensive detail on all aspects of a ship's rigging, including ropes, masts, sails, anchors, and blocks (pulleys), as well as detailed descriptions of seamanship, the practice of working ships, and naval tactics.
The History of the Prevention of Fouling (1952), from the U.S. Naval Institute [Unrestricted PDF] - Explains some of the methods that have attempted to counter biofouling, for both wood and metal hulls.
