When I use Transferwise, I just pay with my debit card. So if I want to send money from my UK account to my US account, I set up the the transfer and then enter my debit card details to send the money - there's a slightly higher fee for paying by card but it's usually not too bad (if I wanted to send £100 to my US account today, the fee would be £1.42 by debit card or £1.13 by bank transfer).
One other option could be to sign up to Revolut, which is a virtual bank account with a pre-paid card (you can get both a physical and a virtual card) - you can add money to the account using your debit card/bank account and then transfer it to another bank account in a different currency... and it has a good exchange rate and low or no fees (sending £100 to my US account via Transferwise today would give me $127.93 plus I'd pay a £1.13 or £1.42 fee, but sending £100 to my US account using Revolut would give me $129.43 with no fee to pay).
Also you can use Revolut to pay for things in various countries in their local currency... it will simply deduct the amount of the transaction from your balance on the card without having to actually exchange any currency or pay any international exchange fees. I find it really handy when I go to the US - I just put money on the card from my UK bank account and then use it to pay for things in US dollars. Or I can exchange that money and transfer it to my US account via the Revolut app. Or I can use Revolut to send money to my friends in other countries without having to use PayPal or Transferwise.
I just tried this. Looked good, was going to cost me 30 pence more than the usual rate for transferring a test 100 pounds. The exchange rate is down to 1.29 now, from 1.34 when I started trying to sort this out. So I got to the "put in my debit card info" and clicked submit. RBS popped up with a "we need to call you to send you a verification code". The only numbers were my UK phone numbers, neither of which are now working numbers. So the transfer failed.
I am going to go into my RBS account and try to put in a US number where I can get calls/texts. But I think that didn't work the last time - I never got any notifications. Either way, I suspect the card may now be locked and a phone call will still be required. (There was a message from Verified by Visa saying not to use the card again until I called.) My current US cellphone plan doesn't do overseas calls, so as soon as I can get into an apartment, have internet again, and can set up my VOIP phone, I can phone them and sort it out. In a couple of weeks. I hope the exchange rate doesn't drop too badly between now and then!
Oh. Well. I have literally nothing else to do for the next 4 days,15 hours, and 46 minutes....
Have just tried the manual transfer and it SEEMS to be working, so fingers are crossed. And I'll only need to do it once more, if so.