How do I update OSM to show that something has moved?

Welcome to the community. It's great that you want to help improve OpenStreetMap! To edit the map you need to have an OSM account. If you don't already have one, you can create one by pressing the "Sign up" button in the upper right corner. Then you can press the "Edit" button next to the OpenStreetMap banner. This will open the iD editor, that provides a user friendly interface to get started with editing. After completing (or skipping) the walkthrough you may start editing. If the fire station isn't yet present in the OSM dataset you can add it by selecting "point", positioning it on the correct place by clicking and dragging it and searching and selecting "Fire Station" in the dropdown. Now you can enter additional information such as name and address into the shown fields.

However you also need to be aware, that OpenStreetMap is only allowed to use data sources that are either license compatible with OSM or have granted OSM access for mapping. I'm not sure that this is the case for london-fire.gov.uk so it's probably best if you only add data, that you can survey either by going there or tracing data from one of the OpenStreetMap-accessible data layers, that you will find in the background pane on the editors right edge.