From what I read, yes, the Bank of England will buy all old coins and notes at face value. A pound is a pound.
Yes, no bank in the US will buy foreign coins and will only deal with paper money if anything. You might look into a Seattle area travel club meeting and see if anyone there wants the coins. You should be able to sell them at face value to someone going to England soon.
Edit: OK, guess I shouldn't rely on memory. You can probably deposit the old pound coins if you have an account depending on your bank in England. So, if you don't have an account and are not planning a visit to Great Britain, best to find a different way to get rid of your old pound coins prior to October.
From the Royal Mint:
What should I do with my existing £1 coins after legal tender status is removed?
Following demonetisation, the current round £1 coin can continue to be deposited into a customer’s account, either business or personal, at most High Street Banks including RBS, NatWest, Ulster, HSBC, Barclays, Lloyds, Santander, Nationwide, Clydesdale, Yorkshire Bank, Halifax, Bank of Scotland and The Post Office.
It may be possible to exchange £1 coins at these banks and the Post Office provided you hold an account with them. Specific arrangements may vary from bank to bank, including deposit limits. It is recommended that you consult with your bank directly.