Me too. I thought it was another British/American difference until I looked at the N-grams. In print, "at high speed" is much more common in both variants.It seems better to me with the "a."
Consider "traveling" instead of "going" in that sentence.
And in American English. I'm surprised that it lags "traveling" by such a great margin.They're probably more common in books and in formal writing but in everyday speech, "going" is perfectly normal in BrE.
And in the above context, I think "fast" would be more common in colloquial English than "at high speed".but in everyday speech, "going" is perfectly normal in BrE.
Something like a train that is variable would be operating at a high speed (out of many possible speeds that could be considered "high.")