Server logs typically have a max size limit so even with the logs you may not recover all the data.
If you have an analytics package installed on the server to read the logs, you may be able read the data in the logs by pointing the settings of the server-based analytics program to the new site log location.
As far as Google Analytics goes, if it was not tracking, you have no data in there for the missing time period. Once the Google Analytics code was removed, it stopped tracking. Google Analytics does not import server log data.