Dont waste your time looking around, firstly check the last modules installed , disable or uninstall them, then try to come back to the upgrade page, you should see all checks marked green. If you recently installed more than one module, its better to tweak around disabling and uninstalling the modules to see which is exhausting the max_execution_time
If no success, then look into the settings of the server max_execution_time to see if it has the right settings. You may need to add it to the php.ini file in the root. it may be worthwhile consulting your host to see if there are any restrictions but that's after all the above has been explored.