Just an update on the situation. Apart from one transfer which mysteriously worked in the middle of all this mess, nothing that we tried did the trick, including trying to use the RSA secure ID device which I paid $25 for and had a really hard time getting them to send to me in the UK. (I'm glad to have it though as I may need it in the future as everything is tightening up so much)
After giving it a break for a good long while, this morning I finally tried a small transfer and paid with my debit card which has worked (YAY!) but I'm just waiting to see if there will be any extra fees at the bank end. They told me there is a 3% fee for an out of country transaction and Transferwise told me that it would be local so I am hopeful but I need to see for myself!
I'll be limited by the daily purchase limit on my card but seeing as I have never made a Transferwise transfer anywhere near that amount, looks like I might me fine from now on, fingers crossed.