This appears to have something to do with your Web Host transfer and DNS/DNS Propagation. The System Info you posted shows that your site was hosted on Freeparking: http://www.freeparking.co.nz/ The new IP address for your website/server is: 65.39.205.57, which is an IP address for this Host: Cogeco Peer1: https://www.cogecopeer1.com/en/ DNS scans show that your Name Server is a Go Daddy Name Server: ns65.domaincontrol.com, which appears to be your Registrar and not your Host server. So either you have not changed your DNS Name Servers/Records to point to your new Host or DNS Progagation has not completed yet.
Apache Modules|Directives|Backward Compatibility(Yes|No)|IfModule(Yes|No): View Visual Test
403:500:500: mod_access_compat and mod_authz_core or mod_rewrite is NOT Loaded
The 403 Status Response means that mod_access_compat is loaded. The first 500 Status Response means that mod_authz_core is not loaded. The second 500 Status Response means that mod_rewrite is NOT loaded. So htaccess files were automatically disabled because the mod_rewrite test results showed a 500 Status Response.
At this point you could go to the Setup Wizard Options tab page and change the Enable|Disable htaccess Files option setting to: htaccess Files Enabled, click the Enable|Disable button and re-run the Wizards, but before doing that go to the BPS System Info page and post your new Apache Modules|Directives|Backward Compatibility(Yes|No)|IfModule(Yes|No): information, which I assume has changed/will be different now. You should also either check your DNS information in your new web host control panel or contact your new web host about your DNS settings.