Are you referring to an Esri Basemap (Esri Basemaps are in Web Mercator)? You can't reproject an Esri basemap.
But, if you are working on ArcGIS Desktop, you could transform the basemap on the fly to a desired Projection system. You could follow these steps.
1. Open a blank Map Document.
2. Add a data (with BNG projection) to the data-frame/ or change the Projection System to BNG from Data-Frame Properties.
3. Add the desired Basemap to ArcMap.
If prompted, choose an appropriate Transformation method.
In case it is a different basemap, please provide further details about it. Also let me know if you are working on a different platform.
Think Location