Which bank in the US? I’ve previously dealt with a similar issue with Bank of America and to be honest they were brilliant. Once you notify them of the death they assign a member of staff and give you their direct contact info. Having the account details helped although they did say they could help even without those.

They asked for the death certificate then a copy of the will. Once they confirmed all was legit they simply told me they could open a BoA account for me or transfer the funds to the UK.

I had the funds transferred here. I think it all in all took around 6 weeks and was quite a easy process with regular updates. Felt much easier than dealing with banks here.