Sure I can connect – I still have the login details – but I have no idea what else I can look for by doing that. There are no known problems with profile saving; no other reports of this behaviour; nothing to look at in terms of console or error status messages…. Everything – points to an environmental issue – i.e., a unique set up of some sort.
My favourite was (and in some ways still is) mod_security. The url passed into the ajax routine are being intercepted and, it looks to me, redirected to a straight http request on the page – hence the return html being the whole page. That could be classic mod_security and yes, I know we have been here before but I personally am still suspicious.
So… @mr-papa – do you have any other ideas not yet investigated? (The WP theme has not been counted out of course).