Linux/Windows are the operating system.
The application program you're using will be something that runs on top of that environment. Since you say the server is webmail.zen.co.uk, I infer that you're using a web browser (Chrome/Firefox/Edge/Opera/etc) to go to that page and log in over the web, and that you then get some sort of timeout, rather than a list of messages or whatever you expect to see at that point.
The implication being that either:
a) Your connection isn't stable enough to complete loading a (possibly relatively large) webpage. This seems unlikely if you're experiencing the same problem from two heterogeneous systems in different locations, and browsing other websites is unaffected.
b) Something is causing the webmail program that runs on Zen's server and turns emails into web pages to fail before it finishes generating a page for you. That Zen support are surprised suggests this isn't some general problem affecting all users (or it is and they're not competent enough to realise it, which seems unlikely).
(I'd suggest that a few thousand emails shouldn't be a problem for a modern mail system, regardless. I have IMAP folders with tens of thousands of emails in them, but I'm not using Zen Webmail, whatever that underlying software might be. 500 is small change. I'd also suggest that exceeding your quota should cause some sort of coherent error message to that effect, rather than just a timeout[1].)
What support are suggesting is that instead of using their webmail system, you try connecting directly to the mail server using a piece of software that runs locally on your own machine, like Thunderbird or Outlook. This would eliminate the webmail system from the equation, and hopefully allow you to access your emails. It might allow you to work around the problem (in the sense of deleting some messages so there are fewer, or perhaps deleting a specific message that is causing some problem for the webmail software), making the webmail system usable again.
(I'd also suggest that using a proper email client like Thunderbird is superior to webmail in most respects - the main advantage of webmail is that you can use it from any random computer without installing anything. Something that most people seem to have forgotten.)
[1] It would be helpful to know if the timeout is coming from your web browser as it fails to load a page, or if the browser is successfully loading a page from the webmail server that contains a timeout message.