Decisions like this can be very difficult. It's nice that patients have options and choices. Without knowing what can and will happen makes it a bit of a guessing game. Let's look at each option.
Giving the bone time to heal can be a good choice. It's non-invasive without the chance of surgical complications. However, if the person is a smoker, has a poor diet/nutrition, or other health issues, delayed healing can occur. Pain and stiffness may keep your mother from getting around during the six to eight weeks it will take to heal. Losing motion in an arthritic knee might cost her some function and independence. Under any of these circumstances it might be best to consider replacement.
At age 83 a patellar arthroplasty (kneecap replacement) will likely last the rest of her natural life. It won't change the underlying arthritis in the joint but it will help keep her moving.
A recent long-term study of patients who had the kneecap replaced suggested older patients do better with a total knee replacement (TKR). With just the kneecap replacement, there are often revision operations needed. With the TKR motion is restored to the entire joint making it possible for the person to remain active.
Given all the factors to consider may help your mother decide what's best for her health and lifestyle. If she tries the wait-and-see approach, she can always have an operation later. If she goes with the TKR her final recovery after rehab will be that much sooner.