To restore everything you've ever backed up using most online backup services, all you need to do is install and then utilize the online backup program on your new computer or device.
It's also possible to use the web-based restoration feature of the service from another computer and pick which of your files you'd want to have restored, even if you haven't purchased a new computer or device yet.
Because you're downloading all of your stuff, it will take a while. Many aspects come into play, but the most crucial is how much data you've backed up and how much information you intend to restore.
Certain cloud backup services allow you to recover your data offline if that's something you're concerned about. To use this service, you must pay a charge and have a storage device supplied to you, such as an external hard drive or a massive flash drive.
One of the rare situations when using a package delivery service to transfer data is likely to be faster than using your internet connection is when you have it mailed to you overnight.