Do I need a TIN to transfer an inheritance from the UK to the US

The word salad they are using is a sign it is a scam.

This line means that you probably don't believe them.

Its pretty much word for word what the email from supposed bank in the UK

You didn't say "my relative X died, and I went to the funeral, the will was read and I heard the lawyer say I inherited £Y. The money has been in my families regular UK bank for months just earning interest and I want to move it". Instead you said:

I have an inheritance In a savings account in my name in the UK

The next steps in the scam will be to get you to pay in advance the COT code fee or other taxes.