In such cases you can always try a manual upgrade. But without terminal console you are a bit limited with hosting environments and for a growing cloud (more data, database upgrade will take longer and longer and you risk to run into timeouts) it’s better to use vserver or other products.
Nextcloud/ownCloud has the largest community in terms of php clouds. There is pydio based on php: https://pydio.com/, for others you probably need a real server to install software (cozy or seafile). If you can transfer files to your webspace via sftp, you can directly mount that storage in Linux/macOS (not sure if there is a free windows version for a drive).
Or if you don’t want to spend a lot of money, you can use Nextcloud on an ARM device (raspberry pi 3 or more powerful: Best cheap hardware to run Nextcloud on?) and use nextcloudplus.com (linux-images based on raspbian which are ready to run Nextcloud).