Hi Martin,
Welcome to SCN!
As you probably know, the SAP service at the OS level runs under the user context SAPService<SID>, i.e. since your SID is HP2, the service user will be SAPServiceHP2. Furthermore, you have a second user, hp2adm, which is the OS-level "administrative" user. hp2adm is intended for user logins on the server console, for the purpose of things like running SAPMMC, etc, whereas SAPServiceHP2 is a system user for running the SAPHP2_00, SAPHP2_01, etc, services, and not intended for direct user login.
So, there is a clever mechanism that the SAP start service uses to take the Windows user environment from hp2adm -- i.e., environment variables like DBMS_TYPE, MSSQL_SCHEMA, PATH, etc -- and 'copy' them to the SAPServiceHP2 environment each time the service is started.
For this to happen, a few things need to be correctly setup for both users. Normally, these two users are domain users, yet you mentioned that hp2adm, at least, is a local user. That can be made to work, but it's not the standard configuration. Normally, the installation is done with a domain administrator account, so that the installer can setup the Active Directory global groups as well as the two domain users. However, you can setup the users and groups in AD ahead of time, and then use a local administrator account to do the install. The installation guide has a sub-chapter on installing without being a domain administrator that details this strategy (and it's the strategy I use in my organization).
So, my guess is something about the setup of the service users and groups is not correctly configured, and it may be related to how you did the install, and to using local users instead of domain users. Using local users is not a supported option, by the way.
To remedy it, I would stop the system, delete (or rename) the local users, then at the domain level create the users again, plus the group SAP_HP2_GlobalAdmin, and add the users to the group. Then at the local level create the groups SAP_LocalAdmin and SAP_HP2_LocalAdmin (these may already exist from your installation), and add the domain group to both of these. If they already existed, you should be good to go, but if not, you'll need to ensure they have privileges to the saploc and sapmnt shares and the \usr\sap folders. Add hp2adm to local Administrators (you don't need to add SAPServiceHP2). Make sure that SAPServiceHP2 has Log on as a service set in local policies (gpedit.msc).
I think that should do it.
Regards,
Matt