You can bid a competitive WOSB set-aside with a pending application. The award itself only goes to a certified WOSB.
FAR 52.219-30(c)(1) solicits offers from certified WOSB concerns or concerns with a pending application showing in Dynamic Small Business Search. FAR 52.219-30(c)(2) is different: sole-source WOSB awards take certified firms only. FAR 52.219-30(c)(4) says any award from that solicitation goes to a certified WOSB.
13 CFR 127.504(a)(3) matches that: at the initial offer that includes price you must either be certified under 13 CFR 127.300, or represent that you submitted a complete application to SBA or an approved third-party certifier and have not received a negative determination. If you become the apparent successful offeror while the application is still open, 13 CFR 127.504(a)(3)(i) requires the contracting officer to notify SBA's Director of Government Contracting, and SBA then prioritizes your file and decides within 15 calendar days of that notice. Under 13 CFR 127.504(a)(3)(ii), if the contracting officer gets no determination in that window, they can treat you as ineligible unless they grant an extension.
13 CFR part 127 does not set a fixed calendar clock for ordinary WOSB application review. What matters for offer eligibility is whether SBA or the certifier has your complete application and you show as pending in DSBS, not how many weeks you have been waiting. Log into the SBA certification portal, confirm the file is complete rather than waiting on missing docs, and check your DSBS record before you submit. I did that once during our staffing WOSB review when a set-aside closed before the certificate posted.