I'm British which I think qualifies me to say it very much was. He had a few inflections of the period, but there's no mistaking it.

I won't tell you what you hear, but his manner of speaking is often held out as an example of Mid-Atlantic diction, albeit an idiosyncratic one—Cary Grant spoke like Cary Grant.

His accent was his native Bristol at base, with Americanisms heaped on top, a Mid-Atlantic destination arrived at from the opposite direction of contemporary American actors. But with his short American a's and clipped cadence, to my ear his delivery often seems closer to that of his American co-stars than to that of his English ones. For example, take a listen to the trailer of The Grass is Greener (1961) [0], in which Grant plays alongside a cast from both sides of the Atlantic. Hear it?

(I'm American and have lived in England, by the way.)

0. https://youtu.be/JpCm22wh6qQ

Just a counterpoint: I have short A's and a clipped cadence. I'm English and have never been to America. (Not that I'm from Bristol either!)

I've never thought of Cary Grant's accent to be anything but English. I don't hear any American in it at all.

I disagree - to me the inflections in his on-screen speech sound predominantly American. (For the record, I'm British too)