Once you have installed and enabled SwiftKey and given it Full Access, you may still need to select SwiftKey as your keyboard when you next go to type.
If a keyboard other than SwiftKey appears, tap the globe icon to the left of the spacebar to navigate through your installed keyboards. Once you get to SwiftKey, just use it! It will remain as the default keyboard until you manually switch to another.
*Please note that some apps do not support the use of third party keyboards, so if SwiftKey is not showing up, this is likely because the app creator or Apple has made requirements to only use the default keyboard on your device.*