At a practical level you need the iPhone. The Apple Watch must be paired to a compatible model iPhone to set it up, and unpairing will dump the watch back to factory condition where the only thing it can do is pair and setup. Once it is paired and the configuration is completed the watch should be capable of acting as a timepiece and tracking heart rate even when its paired iPhone is temporarily out of range.
I will also,draw your attention to the fact that Apple Watch is not an approved device for clinical monitoring. It will spot check (snapshot) heart rate every few minutes but it does not maintain a real time continuous record. While it might detect and alert for an abnormal rate it may also miss short duration anomalies, it may be susceptible to false results if not worn correctly, and the cardiologist may reject the record because it is unfamiliar and unapproved. (On the other hand the cardiologist may be an Apple Watch supporter and be happy with the idea). If your mother is considered to be at risk for abnormal heart rate events Apple Watch may not be the best tool around, and you or your mother should discuss with her doctors before you commit to a decision.