Further to this, I can now confirm that it seems to be region specific. I'm currently logged in using Firefox, with no problems at all, but I had to use a VPN so that it looks as if I'm in Canada (I'm not, I'm in the UK).
I think this confirms that the problem lies with the Trustarc GDPR gizmo, as that seems to try to detect the region of the world you log in from, and if it thinks you're in the EU it will try and pop-up the windows that seems to just hang, and block access to PPRuNe.
So, the workaround, for anyone having this problem, and who is currently located within the EU, is to use a VPN to spoof their location so that they appear to be somewhere else. Don't think it matters much where, as long as it's outside the EU.