Otherwise, what logs does it reveal? If it tells you (e.g.) when services start, you can likely derive the last boot time...
Anyway, is there any name or logo that identifies what software it does use? (Might only be on login page, or on an about/credits page, or in a footer, or maybe only in HTML source).
If so, maybe there's documentation for it that can point towards an answer. Otherwise, if you can't find/enable a way to type commands, and all the logs are fully clogged, I can't think how to obtain the current uptime. Obviously, fixing the time servers and/or firewall to stop the NTP log entries (or even updating config to check less often) might allow that method to work going forward.