There probably won't be any significant long-term effects on climate. A nuclear war will release a huge amount of soot into the atmosphere, which would block out the sun and cooling down the Earth and reducing rainfall. However aerosols do not stay in the atmosphere for long, so pretty soon, the Earth will begin to warm up again. The ozone layer would also be destroyed. However, this is also a short-term effect because without human influence for 500 years, ozone can be naturally reformed.
A possible long-term effect would be the reversal of global warming. Since there would be a lot less greenhouse gas emissions over the next 500 years, as long as plants could grow back fast enough, global warming might be reversed.
3,000 nukes would probably cause a mass extinction. In the years following the nuclear war, huge ash clouds would block sunlight1 causing plants to die off because they cannot photosynthesize. As plants begin to die off, so will animals. The largest organisms would suffer the most because they require the most energy to function. However, some organisms might be able to adapt to the new environment and they will be the ones who repopulate Earth over the next 500 years.
You really can't predict what the environment will be like in 500 years. Most organisms will probability need adaptations for a harsher and more radioactive environment. 500 years isn't enough time fore Earth's environment to recover, since the nukes would have created patches of lifeless rock, and succession takes hundreds of years.
p.s. This is my first time answering, and I would really appreciate any comments.