The maximum load of the elevator doesn't depend on your power technology, but your material technology.
- the maximum load of a lifter on your space elevator is determined by the thickness of the lower end of the cable.
- the thickness of the lower end of the cable will be kept as low as possible to reduce the weight of the entire elevator, and hence the difficulty in building the thing in the first place.
- the thicker the bottom end of the cable, the fatter the middle of the cable, and the heavier and more complex to make the whole elevator is
Really, anything from 20 tonnes or a thousands tonnes or more is fine... handwave in whatever you want to suit your story needs. Lighter loads for lower tech levels, partially constructed elevators with only a thin cable that's still being expanded, higher gravity worlds, etc.
The speed of your elevator is is limited partially by how it is attached to the cable, and partially by how it gets power. Elevators are more efficient than rockets, but there's still a lot of energy required to lift a load up to the geosynchronous point.
Specific orbital energy is the number to care about here. The important parameters are $R$ the radius of the planet you are lifting from, $a$ the radius of a geostationary orbit, $M$ the mass of the planet and $G$ the gravitational constant. $\mu$ is the standard gravitational parameter, $GM$. The amount of energy you have to expend to raise the lifter is defined as $$\epsilon_a = -{\mu \over 2a} + {\mu \over R }$$
For Earth, you'll have an $\epsilon_a$ of ~57.77MJ/kg. A hundred tonne lifter would need ~5.777TJ. If your lifter managed an average speed of 600km/h, you'd make the trip from the surface to orbit in a little under 60 hours, and need a steady power supply of about 27 megawatts sustained for the whole journey.
If fusion power is effective, compact and safe, you could fit a mini reactor to the lifter. Given the weight requirements, this seems unlikely unless you have a really substantial cable and a very large lifter of a few thousand tonnes or more. If fusion reactors are big and heavy and awkward (which is likely), you might keep them on the ground (or in orbit) and use laser or microwave beamed power to drive the lifters motors.
If you wanted a faster trip, you might be able to handwave one in given suitable power supplies, though cable friction is going to be a problem. Some kind of fancy superconducting magnetically levitated linear motor might work, but all that extra weight on the cable has to be held up by something...
Please also remember that you get one lifter per cable. if you want multiple lifters, you'll have difficulty fitting them past each other, even if the cable was very large indeed. Here's a reasonable artist's impression of a lifter. See how it encompasses the cable, and is quite broad in order to accomodate beamed power collection arrays:
(image credit Liftport)
I'll spare you the gory details of elevator cable width, but if anyone really wants me to walk you through it and doesn't fancy reading eg. The physics of the space elevator to work it out themselves, do ask.
