I’m saving up Euros, and keeping them in Fineco, an Italian bank that offers its services to UK customers and in English.
They also offer a USD account, however, I don’t think it is as good as the Euro account as they don’t offer a debit card for it, and as far as I’m aware, they don’t support ACH transfers.
HSBC and Barclays both offer USD accounts. Again, no debit card, and bank transfers cost $6 each. I believe they do support ACH transfers.
Maybe other people can offer suggestions, as I haven’t really looked at the USD market.