At any point in time you can change who on your team is assigned an alert by clicking on the current assignee and choosing a different name from the list, or just assigning it to yourself (you will always be the first one in the list of users).
Only users whose roles include edit alerts rights can be assigned to alerts. See user access management for details on role permissions.
⌨️ Keyboard shortcuts
When in Alert page:
click A to prompt Assignee menu
use arrows to navigate through users
click Enter to select the Assignee