In the U.S., there is no expression for that. We usually say "no sugar" or "with nothing in it." Most Americans don't say "neat," but some do, especially Anglophiles.
It's fine to say "straight" or "straight up" ordering coffee. The server will understand that you're making a little joke and that you don't want anything in your coffee.
Some people in the U.S. say "regular" when they order coffee, but it's dangerous, because "regular" means different things in different parts of the country. In some places it means to put nothing in it. In others, it means to add milk and sugar. In others, it's half-and-half and sugar. In others, it's just sugar.
So here in the U.S., it's safest and most common to say "coffee with nothing in it."
I don't know what they say in England.
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