My question in essence is, once the user is authorised with the API, how should this authorisation be persisted on the front-end?
In a RESTful API, after authentication you will get a access token. Your front-end have to send this access token on every request to API. Backend(API) will validate this access token and provide infomation.
If you are building a web api consumer, you can have a fallback cookie option (so users don't have to authenticate every time they close browser tab). On others api consumer (mobile apps for example) you can store the access token in someplace safe