By my rough calculations, a fall to the center of the Earth will raise the temperature of water by some $7,200 K$. This is extremely hot, but I believe it falls short of a full plasma ionization. Additionally, I would put the pressure at the center somewhere around $30 GPa$. That's a "G", not an "M". I can't even look up properties because these conditions are so extraordinary. However, I believe that we will be looking at a supercritcal fluid either way.
I take this question to refer to the "fast" event where water is let in. If it was left to reach thermal equilibrium over a billion years, the answer might be slightly different.
Regarding the fall itself, I would go ahead and assume that the fluid quickly reaches its own form of terminal velocity. Even with a 12 meter pipe, and viscosity reduction due to phase transition, the Earth is huge relative to the pipe. Frankly, that means that there's little kinetic energy by the time it reaches the center as opposed to thermal energy. In order for it to make it back up to the other side of the hole, it has to be pushed through by the back-pressure, which is also enormous. This process will take much longer than the time required to complete half of an Earth orbit (about 40 minutes), which is what would be involved for a free-fall. Maybe a few hours, it's an interesting question.
Also keep in mind that energy is liberated in the process. You stored up a lot of energy by building a vacuum tube through the Earth. As the water falls, that gravitational potential energy is converted into thermal energy in the fluid.
Play this movie out, and you realize that the fluid won't stop moving at the center of the Earth... or the surface on the other side. "But wait!" you protest, this like like a pendulum, it has to climb the same gravity well that it fell through. Yes, but its density changed due to the heating. That means that the falling fluid is much more dense than the climbing fluid. The climbing fluid will undergo a relatively isentropic expansion process, but its enthalpy will always end at a higher value than the original inflow (because the only net heat flow is frictional heating).
Because of this, a geyser would sprout at the land-based entrance to the tube. It should probably release a volume similar to the total volume of the tube, and it will be at massively super-critical temperatures. This will be very dangerous, but I predict that it'll fall short of climatic scale impacts.
In fact, the geyser would eventually stop, because all the energy is coming from liberating a finite gravitational potential energy source. Then there would be some near-term equilibrium state where the tube's fluid is significantly heated, but the driving force has basically puttered out. As this process happens, the flow rate goes down, and so does the thermal energy of the geyser. Eventually it might just be regular steam that it belches out, and then water, although very hot water. Toward the very end of this process, it'll just be lukewarm water pushed out by a more mild driving force. However, depending on the elevation difference, this could continue to pump a large volume, larger than the volume of the pipe I think.