You can say
The USA and the UK have been at war once since 1786.
if by 'at war' you mean 'in a state of warfare (against each other)'. But I find the use of have been at war a bit unnatural.
As far as using the simple past, you can say:
The UK and the USA were at war (against each other) once since 1786.
If you want to talk about going to war against each other you can say
The USA and the UK have gone to war (against each other) once since 1786.
1786 is the date that the UK recognized the USA as an independent country. The one war between the two nations since 1786 is the War of 1812. The conflict that began in 1775 was a (successful) war of independence or revolutionary war, fought between the usurpers of liberty, the Empire versus the so-called Rebels, kind of like in Star Wars, with William Howe as Darth Vader, Benedict Arnold as Anakin Skywalker, and King George III as C3PO.