Say for example you got a bank account at [bank x] in the UK, denominated in GBP. Now you'd ask [bank x] to set up EUR denominated account. Then use transferwise and transfer out of your GBP to transferwise into your EUR account. Then use those EUR to pay and avoid the double bid/offer.

//edit: to clarify: you would send the money to [bitcoinexchange], not transferwise. so it would be in your name.