Confirming the applicant's current citizenship status itself should be quick, based on existing passport or citizenship documents. The open-ended part is how long a pending citizenship application might take, which depends on the applicant's own process rather than anything TN-specific.
Separate the quick check from the open-ended wait
Get the citizenship documentation reviewed right away, since that answer does not require waiting on anything else. If citizenship is pending, ask for the current status of that specific application, since it drives whether the household should plan around a future TN route or explore something else now. Do not let the job offer's own timeline pressure a decision before this threshold question is answered.
Split the question into a quick check and an open-ended wait, and treat them differently. Reviewing existing passport or citizenship documents can be done immediately and requires nothing from anyone else, so do it first. How long a pending citizenship application takes belongs to a different process entirely and cannot be accelerated by any immigration preparation, so ask for its current stage and plan around the answer rather than around a hope.
Do not let the employer's own deadline pressure a decision on the threshold question, because a route pursued without qualifying citizenship does not become available through urgency. Set a review date and keep the employer informed of the actual position.