So after spending an hour or so attempting to connect to a SQL 2008 R2 Reporting Services instance, which was hosted on the same server as an existing SQL Server 2012 Reporting Server Instance, I discovered that no matter which combination of Server name / Instance name or even SQL Alias I used, I simply was not able to make a connection to the 2008 Reporting Services through the 2008 Configuration manager.
I was able to make a connection to the 2012 Reporting Service via the 2012 Reporting Service Configuration manager though. Not what I needed.
Stopping the SQL 2012 RS service did not resolve, and the only resolution I could find was to uninstall SQL 2012 Reporting Services via Add/Remove Programs from the Control Panel – which fortunately was installed but not needed on this occasion.
Conclusion: SQL 2008 and SQL 2012 do not play nicely together on the same server, regardless of whether they are named instances or not :/