Existing electronic health record (EHR) systems already have to deal with this.

The ones I am most familiar with have on-site data centers, so it's not purely in "the cloud" but you can have webapps running from on-site servers and call them "cloud" systems if you want.

The existing systems have ways for printing out important data from a redundant computer system to hand out paper copies if the user workstations, network, or servers are down. I don't know what they would do if there was no electricity anywhere, but I assume hospitals could use their backup generators for this since medical equipment also needs electricity.