Here is the UK - US tax treaty. Scroll through it until you reach article 17.
The US can tax the pension of a person who is a resident of the US unless the pension is exempt in the country of origin, since the person resides in the U.S. However if the pension is tax-exempt in the UK, it is also tax-exempt in the U.S.
And you are correct, no forms are required in this situation.
@MovedtoNZ
