It's an interesting question that I don't think gets much attention, but it's one that really needs just as much planning as the part of disaster recovery that includes keeping your data safe. The answer depends on the nature of the disaster. In situations where your data center is intact, such as after a major power outage that exceeds the duration of your emergency generators, then it may be that all you have to do is update your data from what's in the cloud and start running again.
But chances are that a disaster requiring you to make use of the data stored in the cloud will also damage or destroy your data center. What do you do then? If you're asking this question after the disaster has already happened, you'd better hope that your data center environment isn't too complex, and that you can find a work area and computers to support your needs. A better time to ask this question, however, is at the same time you're doing the rest of your planning.
There are several options. If your company has more than one office and the offices are in geographically dispersed locations, perhaps you can outfit each office to act as a backup location for one that's out of operation. This may require some extra resources in your data centers, but you can make use of that for other projects during non-disaster times.
You can, of course, contract for a site in advance, which means that you'll have a data center waiting for you, and all you have to do is reload your data at the new location. But perhaps the best solution I've seen is to contract with the same cloud provider to create a cloud-based backup of your data center. Not only will your data be safe, but you can run your applications in the cloud and access them remotely. Your cloud provider will charge for the availability, but it'll cost a lot less than keeping a full-time backup site.
The real key, however, is planning or training. Once you decide on where your data will be stored, and how it will be used until your data center is available again, you have to plan every detail of how the transition will take place. And then you need to practice. Frequently.
The only way you'll be able to know for sure that your current environment can make the transition to a new location whether it's at another office or in the cloud, is to try it out. The process of practicing will also make any gaps in your planning obvious, and it will allow you to revise your plan so that it will actually work. And once you've done that, you need to practice it again.