In order to answer this question we need to know whether an Unlawful Detainer action was filed in State court prior to your filing the BK, and, if so, the status of that action on the date you filed.
Your timeline could be anywhere from 10 days to a couple months.
Their motion for relief from the stay, if granted, will allow them to proceed with the Unlawful Detainer (eviction) in the state court. Like my colleague stated, we must know the status of that case, prior to the filing of the bankruptcy. You should consult with an attorney to go over the facts of your case and advise you of your options.
I personally don't think it's a good idea to file bk just to stay in a rental a few more days. Filing for bk is something special you do to restart your life, but here, it sounds like you're going to come out of bk owing $ to your landlord.
The above is general legal and business analysis. It is not "legal advise" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here. Please visit my web site: www.avanesianlaw.com for more information about my services.