What would happen if the US promised not to use the death penalty......?

It wouldn't be because for the prosecutor to push for the death penalty after a sovereign promise was made by the US executive branch not to do so would be an abuse of process. The President of the United States, though his power of pardon, has the ability to commute or alter any punishment or sentence, even before it has been handed down.

So if the US executive branch went back on their word and ordered the US attorney to push for the death penalty in the case, the judge would rule it an abuse of process and exclude it as a possible sentence. If the judge decided that, contrary to what the executive branch had promised, he would allow the death penalty to be considered or even imposed as the penalty, then the president would simply commute the sentence from death to imprisonment for a term of years.

If the US executive branch and judge conspired to allow the death penalty to go forward for consideration, then it would be overturned in the relevant circuit court of appeals. There are sufficient safeguards in the US legal system to prevent such a thing.

But ultimately, if we go with a fictional scenario where somehow none of that was the case, then yes the UK's only recourse would be diplomatic.

Would that possibly mean a trade embargo, ie, we don't accept imports of any of their goods they sell to us?

I fear that the tough as nails Donald Trump would be the most likely to allow something to be broken as significant as promising not to use the death penalty.

That wouldn't happen. We'd let the person be executed before allowing such a fundamental breach of relations with the United States to occur. In any case, during the 12 or so years that it takes for someone to exhaust their various avenues for appeal while they are on death row, the UK could bring discreet pressure to bear on the president to live up to the promises made.

It's not plausible that the US would ever resile from such a promise, it's not in their interests to do so. It would damage the United States far more than what they would "get" out of executing someone rather than imprisoning them for life. The United States government benefits when other governments perceive that it is good to its word and lives up to its promises; it is easier to convince other nations to do what you want them to do when they believe the promises you make. I can't think of any situation in the past where the United States has resiled on a prosecution/extradition agreement of that sort.

And as I said, I do not think the US federal court system would allow it to occur anyway; it's an abuse of process.

Let's say the UK did still have the death penalty, but only used it very sparingly, do you think the process you just mentioned above would be the same or similar?