Thanks @phelings. I've had a bit more success since I last posted but I also have days where I can only get 1 episode to download before it does the "flick to the account screen/log me out" error I mentioned above.
I'm definitely having more success without using the VPN and sometimes manage to get 4 or 5 episodes in a row before the error occurs.
I've also noticed that if I get the error, pause all downloads, close the ITVX browser window and then go back into the ITVX app, it then analyses the TV series that is on the front page of ITVX (currently Vera). If I wait for it to finish analysing, close the analysis window and then start my ITVX queue again, it seems to work for more episodes in a row before I get the error again (although I might just be fooling myself with what feels a bit like a superstition).
I can see above that @north_east21 mentioned they were having the same error, so if you get chance and you haven't already, could you open a ticket about the issue via the Get Support button in the Member Center. Assuming there isn't an update in this week's release, I'm going to send another log to the support in the hope that the more reports they get about the error, the more chance we have of them trying to figure out a fix.
Thanks.