I've logged this with support, but have done some further research into this whilst waiting for their reply.
It appears to be an issue with Kentico using cmsctx URLs instead of proper URLs in the iframe. If I compare the code in the local environment to the test one, the local has a proper URL for images, scripts and stylesheets, the test has cmsctx URLs.
A good example is the header image:
Local: /img/header/logo-home.png
Test: /cmsctx/pm/bd6edaaf-ff69-4e53-a252-6a1d567c849e/culture/en-GB/wg/bf922dd8-6c74-4dc6-a435-cca41fd94b27/readonly/0/ea/1/h/54116ffcb9512489d8ba75a9adadfc52e3963281189fd4b2a1e0ce296034e37f/-/TSCP/img/header/logo-contact.png?uh=b3c192a550ffebab0f664d7d8496201d16227d5202f1ede9bd142741fee49a3f&administrationurl=https%3A%2F%2Fdevserv04.tameside.gov.uk%2FTSCPAdmin
If I change the URL in dev tools to '/TSCP/img/header/logo-contact.png', which is the actual URL, then it loads as expected, this is the same for the css and js files, so it seems that Kentico is using the wrong URLs for the files. I'm not sure if this sheds anymore light on the problem.