If I read your question correctly, living donors can be used. There are associated risks and the donor can develop diabetes themselves. I suspect they used deceased donors due to the complication possible for live donors making it harder to get ethics approval.
Dr Coates mentioned in the Webex that the vascularised bed created with BTM can wait several weeks until donor islet cells are available to be immediately transplanted. The holy grail would be to use stem cell derived islets that are autologous (from the patient) but that technology is still maturing. It might be that these can be grown while BTM is introduced to the patient.
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