Is it obligatory for someone who has completed the circumambulation (Tawaf) but was unable to perform the ritual walking (Sa'i) to repeat it if they start the 'Umrah anew after a period of rest?

Consecutiveness between the circumambulation (Tawaf) and the ritual running (Sa'i) is a Sunnah, not a condition for the validity of 'Umrah. So, if there is a short or long break between them, it is permissible, and the Sa'i is valid, as long as no standing (wuquf) intervenes. If standing does intervene, then the Sa'i must be performed after the Tawaf Al-Ifadah. Your 'Umrah is valid if you recommenced the Tawaf. Your first Tawaf, if completed, is sufficient, and the second one is supererogatory.