The failover is done by a separate controlling server so it will not be affected by the hypervisor's failure. Also the actual virtual machine image isn't on the troubled hypervisor, it is on a SAN disk raid so it can be migrated to a functioning hypervisor even if the original one isn't responding. Failure though is possible if the SAN isn't responding. Also, migration may be delayed depending on several factors.
For example windows servers will need time to reboot in what is called cold migration. This is a limitation of the OS not the cloud. Most if not all unix-based OS's will be able to hot migrate which means access is only temporarily turned off then on. Sometimes if the virtual machine's OS crashed, when restarting that OS may run a check disk and take longer to start up.