Hi,
I have developed software for over 20 years, and I have lots of experience with most web-related technologies, both server-side and client-side, including HTML, web servers, php, SQL databases, etc.
There are two possible approaches to your problem: one is to optimise the database and/or database query so that the report takes less time to execute. This depends a lot on the amount of data, the database schema, your current server configuration, and more. If you could provide details of all this, that would help a lot in providing a more accurate cost and delay. I will need access to the database and the source code of the scripts to address these issues.
The other approach is to simply increase the timeout so that your report gets displayed even if it takes longer. There are several different places where the timeout could occur (web server, php...), I will need access to the configuration of both to make changes there.
There may be issues if you are using a low-end hosting solution (some form of shared hosting in particular) as you may not have access to all of the configuration. In that case, I will recommend migrating to a different hosting solution.
Let me know if you can clarify the above points or if you have any questions.
Jacques.