When you check the web developer console, you'll notice the error
TypeError: $.cookie is not a function
This is most likely caused by the missing file
http://www.imperialcharity.co.uk/new/js/jquery.cookie.js
In case this error is fixed by uploading the missing file, it's possible that everything will work fine.
To add more detail - the initial server response header for this file is the status code 406 - Not acceptable
(for detailed explanation you can check What is "406-Not Acceptable Response" in HTTP?), but as the server response is "An appropriate representation of the requested resource /new/js/jquery.cookie.js could not be found on this server.
" , uploading the file might already fix the problem.
Not relevant to the problem but while checking the net console I just noticed that there's also an image missing - http://www.imperialcharity.co.uk/new/css/loader.png
Update: The mentioned error 406 with the jquery.cookie.js is still there. I just checked the DesignChemical accordion plugin with the jquery.cookie.js that comes together with it and it looks fine. I would suggest to rename the file - e.g. to cookie.js instead of jquery.cookie.js and to change the include accordingly - but as I understand you already tried something like that.
To make it easier for anyone trying to help you to solve this issue, it would be good if you could update/add approaches already tried in your question as comments could get overlooked. Also it would be good to know what happened when you renamed the file - if not already done, open web developer tools and check the net panel. Just in case you're unsure about that, e.g. for Firefox it looks like this: https://developer.mozilla.org/en-US/docs/Tools/Network_Monitor , for Chrome like that: https://developer.chrome.com/devtools/docs/network. It would be good to know if in case of renaming the file the error is still the same.
Just to exclude following possibility, you can download the uploaded file, open it and check if it looks o.k. / the same like the file working for you locally. Maybe it happened that the file wasn't uploaded completely and is broken.
And as a last guess you can check if the file permissions for the jqery.cookie.js are the same as for the other js files. Though in case of different permissions the error would be different (e.g. 403 Authorization required
instead of 406
) one never knows how the server was configured. All the other js files in the folder are accessible, e.g. http://www.imperialcharity.co.uk/new/js/jquery.dcjqaccordion.2.7.min.js, only http://www.imperialcharity.co.uk/new/js/jquery.cookie.js behaves uncooperative.