Publish details visible only in edit mode

Please help, i need to publish a map on the institutional website of an hospital.

I need to make visible for the public in this planimetry:

all the details (code of the pavillions, for example A, A1, B1, B2 B3, B4 etc) filled for the same building that i can see in edit mode here:

Many thanks.

I wonder if this is just a caching issue? I see the building names on the map.

I assume you mean that the ref=“something” that are added to the building:parts?
Those are not rendered on the available layers at osm.org, unfortunately.

You probably want to use something like Maperitive, or some other Rendering or Export solution
(or maybe even GIS software), that lets you pick exactly which objects and tag information you want to include in your map and in what style.