It will depend on the type of Watchlist "flavour" you have. If you manage watchlist via the User Interface and not via the API you can click on the edit option in the WatchList section of the Admin interface.
In the event you manage your own WatchList using the API only, there is currently no API call to retrieve the content of a specific watchlist.
This is something which could be considered as an improvement for the future. If you want to raise an enhancement request please use the following link:
Your request will then be prioritised.
Thanks, Léo, that answers my question. We use the API.
I think an API endpoint would be very helpful, both from a development perspective but also ongoing maintenance. An alternate stopgap would be to allow Watchlist editing from the API and console at the same time. That is probably much easier said than done though.