are we talking a couple weeks, months, years?"
It all depends....
Some of the foothills terrain just has brush burned off and the trails aren't really affected by the fire. There will be steeper terrain in places along with trees that will be more affected. So unless the area is closed you could realistically ride much of it soon after. I've ridden through (smaller) burn areas a day or two after.
Closures are anyone's guess. Given the size of the fire I'd expect some closures, but it could be a patchwork as the various open spaces around are managed by a bunch of different land managers. I'd hope any closures would just last a few weeks until hazards can be assessed and dealt with (ie burned trees) but some areas could be closed through spring to let some vegetation grow in to help prevent social trails or stabilize the soil. No reason to keep them closed longer than that.
It will be a lot worse for the Eaton Fire. The Angeles NF always has closures after fires because of the steep terrain and tree canopy. Some of those have gone for years since there can be damage not just from the fire but from storms that follow. It takes time for the slopes to stabilize which usually includes having vegetation grow back. And then the trails have to be rebuilt as well.
Mount Wilson Bicycling Association and Lowelifes Respectable Citizens Club are still working on trails burned in the 2020 Bobcat Fire. The ANF has slowly opened areas to the public as the trails get repaired. MWBA is still working on Valley Forge while Lowelifes have been working on Sturtevant and the Gabrielino between Sturtevant and Newcomb Pass. There will be more pressure to re-open trails on the front side. But where damage is severe it could take a long time.
We need rain but hopefully there are no heavy storms this winter. That is what did the most damage following the Station Fire in 2009. The Gabrielino Trail from Switzers down to JPL wasn't reopened until 2018. And Condor Peak in 2022. MWBA and Lowelifes are motivated and highly organized now, and won't let the trails languish for that long. Please support both groups! Trail days are rewarding and fun, and if you can't make it to those donations go a long way as well.