How much longer does Earth have until the core turns solid?
Does global warming change these estimates at all?
Global warming has to do with the surface only, and at best involves changes of 20 degrees at the outside extreme, in comparison to the earth's core, which is as hot as the surface of the sun.
For complete accuracy, and to reflect what a commenter has pointed out, the inner core is solid already, but this is because of the extremely high pressure of the overlying layers of the outer core (which IS liquid), and the mantle. See the Wikipedia articles concerning the Inner Core, and the Outer Core. Note that it is the outer core which creates the earth's magnetic field.
The answer is that the earth's core will never be solid. And I do mean NEVER. Now, that being said, there is only one way it could ever happen and that is if the earth happened to get thrown out of its orbit to become a nomad planet. Then it might have time for its core to cool.
The reason I say this is because it will take longer for the earth's core to turn solid than it will take for the sun to run out of nuclear fuel and expand to engulf the earth. At that point, the earth will be vaporized as it spirals out of its orbit into the sun. The core would soon turn into incandescent gas. This will occur something like 4 to 5 billion years from now.
If by some chance the earth were to become a nomad planet, free to cool in its own good time, then it would take a long long time. See Energetics of the Earth by John Verhoogen, available online via Google Books.
The main factor slowing down the cooling is radioactive decay of long living atoms, namely Uranium-238, Uranium-235, Thorium-232, and Potassium-40, with half-lives of roughly 4.47 billion years, 704 million years, 14.1 billion years, and 1.28 billion years, respectively. From the half-lives of these isotopes and a comparison with the age of Earth, you can see that internal heat production via radioactive decay will likely persist at near current levels for quite some time to come. Verhoogen gives 5000 K as the core temperature now, and a 250 K cooling since the formation of the Solar System, 4.5 billion years ago. If it really does cool at that rate (55 degrees per billion years), it would take something like 91 billion years to cool to 0 Kelvin.
But don't worry, it won't happen, as I said.
Edited to add detail