Hi, does your British husband actually have a bank account in the UK? If he does he can, if he wants, merely pay Traveller's Cheques into his bank account.
Those Traveller's Cheques, will they be denominated in Pounds Sterling, or in US$. If in US$, no problem, the bank would be able to exchange them into Sterling, and then credit his bank account with the sterling amount.
Also, if he does have a bank account in the UK, why not simply transfer money into that UK bank account? That is, go into your Bank of America branch, fill in a form, giving details of his UK bank account, and obviously how much you want to transfer, and let the transfer happen on a bank-to-bank basis.
Or indeed, if you have internet banking, you might be able to initiate that transfer online.
Would he be able to go into a Barclay's branch and withdraw straight from our accounts with a teller in any amount?
Withdraw from your BofA account in the US? Certainly not.
If he just took over a traveler's cheque for say, £1500, would he be able to cash it or something to use toward the deposit and rent?
Obviously yes, but would he want to be carrying all that cash?
Finally, Western Union! Not very cheap, but it is almost instant ..... well just a few minutes. There are many "branches" of Western Union in the UK.