This is actually a complicated question/answer - it depends on what aspect of space weather you are concerned about.

Some effects, such at those related to the x-rays from solar flares, travel at the speed of light and reach the Earth's atmosphere in about 8 minutes. The effects that x-rays have on the atmosphere affect radio signals and the changes can actually last a while.

Other effects, such as the particles that are accelerated near the Sun (usually by shocks created by the huge explosion of material we call a coronal mass ejection, or CME) take a bit longer to get to Earth. Depending on the energy of the particles they can arrive within hours or a day. These affect astronauts as well as instruments on satellites and the danger can last for days.

Then the CME itself can travel to Earth and hit the magnetic field that Earth has around it. This causes a whole range of effects that also last for days and cause problems for spacecraft near the Earth as well as have effects on the Earth, for instance in power lines etc. How long it takes for the CME to arrive depends on the speed it is traveling and can take anywhere from about 1-4 days.