Hi Giulio,
Based on your description, I tested the scenario as yours. After applying SQL Server 2012 SP2, I reboot the server , then restart all the associated SQL Server services that use NT SERVICE\<SERVICENAME> accounts successfully.
According to the error message, it might be caused by that NT SERVICE\<SERVICENAME> accounts don’t have sufficient permissions to access the SQL Server services installation folders. You can post detailed information in the SQL Server Errorlog file
for analysis.
In addition, Local System is a very high-privileged built-in account. It has extensive privileges on the local system and acts as the computer on the network. And it is not recommend to use local system account for running SQL Server services. For more details
about configuring SQL Server Service accounts, you can review the following links.
Configure Windows Service Accounts and Permissions:
http://msdn.microsoft.com/EN-US/library/ms143504.aspx#Windows
Best Practices For Using SQL Server Service Accounts:
http://blogs.technet.com/b/canitpro/archive/2012/02/08/the-sql-guy-post-15-best-practices-for-using-sql-server-service-accounts.aspx
Thanks,
Lydia Zhang