Thanks both for your suggestions. I appreciate the help.
I think that both, when looked at closely, are one and the same.
I did use the Remove Library function on the Calibre Library icon/button.
What happens is that, once the target library has been selected from the list, a dialogue box appears virtually immediately, with options to proceed or cancel. However, once the request to remove the library is confirmed, Calibre then spends just over a minute seemingly looking for the darn thing, with the hour glass spinning.
Because the libraries in question are on a server which is no longer connected, it never finds it, and eventually gives up trying, and goes ahead and removes it from Calibre anyway.
Removing a library which is accessible is virtually instantaneous. It is only in a case like mine, where it is not, that the long delay ensues. I had hoped that there might be an option to remove a library without the delay, but apparently not.
Pernaps, Kovid, you may wish to check whether that delay is actually serving a useful purpose.
In the scheme of things, I've probably spent more time typing this lengthy explanation, than it took me to remove my libraries at just over a minute each.
Thanks again.