How do I choose the right replacement fuse for my multimeter?
The impact would be that the fuse should be able to break the circuit if you connect the multimeter to a 1 kV voltage source.

Basically for fuses, the voltage rating tells you which is the highest voltage the fuse will break. The higher the rating is, the safer you are. If, for example, you connect the fuse to a voltage higher than its rating, and the fuse opens, it could fail to break the connection in the rated time period.

For example, a plasma arc may appear at the location the wire inside the fuse breaks that will remain conductive for some time. For that reason, high voltage fuses often have the wire going through some sort of a labyrinth, in order to provide the long distance in hope that the plasma arc will extinguish inside of the fuse.

The current rating tells you how much current can go through the fuse without it breaking. In general, you should substitute old fuse with a fuse that has same rating. If that is not possible, go for one with a lower rating. This way, you'll prevent the instrument from using its full current range, but you'll avoid having some part of the instrument other than a fuse from acting as a fuse.

As for the speed rating, it's always best to go with the same rating as on the old fuse. Fast fuses will react to overcurrent very quickly, while the slow fuses will allow overcurrent for some time. Some devices that have large in-rush currents may use slow fuses, so the fuse doesn't trip when the device is turned on. On the other hand, using a slow fuse on a device that expects a fast fuse will bring the danger of having some other part of the device act as a fuse.