You can access map services supplied by the Ordnance Survey directly in ArcGIS Online by using the OS Maps API.
The API lets users access a variety of map styles and projections. The styles range from being ideal for outdoor use, to being optimised for using on a tablet at night. Scale can vary too, from an overview of Great Britain as a whole, right down to the level of detail needed to identify a single building.
Once you have signed up with OS and obtained your API key from them, the process to consume the map services in ArcGIS Online is simple.
Just go to the map viewer and select "Add Layer from Web". Choose "A WMTS OGC Web Service" and enter the URL https://api2.ordnancesurvey.co.uk/mapping_api/v1/service/wmts?key= plus your key
Make sure you select "Use as Basemap". Click 'Add Layer' and you are good to go. You can now use these basemaps in you applications within your ArcGIS Organisation.
Full details of the OS Maps API and the other APIs that the OS offer can be found on their developer portal.