Because doubt is an old word (the OED attests it from 1225) there are a number of nuances of meaning. In summary, with my numbers not OED's:

1. A subjective state of uncertainty about the truth or reality of something;
2. (plural) A feeling of uncertainty as to something;
3. The condition of being objectively uncertain;
4. A state of affairs which gives rise to uncertainty;
5. A matter or point involved in uncertainty; a difficulty (Obsolete);
6. Apprehension, dread, fear (Obsolete);
7. A thing to be dreaded; danger, risk (Obsolete).

As a singular noun, it is decidedly abstract and refers to a state of being. Thus one can be in doubt about something. As a mass noun referring to a state of being, doubt would not normally be pluralised.

When it refers to a particular feeling of uncertainty it is pluralised. Thus if one has that feeling, one has doubts. One does not "have a doubt" because the singular doubt refers to a state of being; senses 5, 6 and 7 which refer to particular things are obsolete in Standard English.

The uses which are obsolete in Standard English may have survived or been resurrected in other dialects. The ELU references say that it does occur in Indian English.

It doesn't grate on my American ear. Though I'll concede that I have doubts sounds better, I had to think about it.

I checked my usual sources: Ngrams and Google books, and concluded:

• The (perhaps more technically correct) I have doubts is more common, though both can be found.
• The Ngram for British English has a perceptibly flatter line, so it may be a bit more common in the US than in the UK.
• I did find this except in parliamentary debates (although the excerpt does not indicate who is speaking):

I wish to see legislation that works. At the moment I have a doubt about its workability. I have asked this question before to the noble Baroness, Lady Scotland .. and I did not receive an answer.

(maybe she snubbed him because he said "I have a doubt" – who knows?)

_{Source: The Parliamentary Debates (Hansard): Official Report, Volume 674 Great Britain. Parliament. House of Lords, 2005}

EDIT:

(to FumbleFingers and the downvoter, who may not be one in the same)

Okay, I didn't want to have to do this, as I included some links for the curious to explore. However, if more samples are needed to thwart the downvoting, so be it:

I have a doubt about this story. Whereas Holt has displaced the nationalist preoccupation with whether freedom was given or taken he nevertheless reproduces a story about the ultimate failure of the liberal emancipation project to confer substantive freedoms.

– quote taken from Refashioning Futures: Criticism after Postcoloniality, by David Scott, 1999. Yes, Scott did write a book covering Mother Teresa – maybe that's where he picked up this horrible English phrase? – but he was born in Ohio, attended seminary in Pennsylvania, and he recently served a two-year stint as editor in chief of the Catholic News Agency.

You owe it not to modern science, but to ancient superstition. Had they not obligingly provided the commandment, it were vain for you to provide the camera. Besides, I have a doubt about this scientific superiority of fire-arms...

– quote taken from The Scientific Mind, a 1927 short story by G.K. Chesterton. Yes, in the excerpt, the cook is talking to the professor, not the other way around, but I don't think Chesterton is trying to make the cook sound ignorant or uneducated.

It isn't always possible to know why a given sentence in your overview isn't effective, but if you suspect a given sentence isn't strong or if you have a doubt about whether a given idea is a plus, rethink and revise the material in question.

– quote taken from The Art of the Book Proposal, described by the publisher as "an expert's guide through the elements of a nonfiction book proposal." It was written by Eric Maisel, an American author who has written more than 40 books.

There is nothing that intercessory prayer cannot do. Oh, believer, you have a might engine in your hand, so use it. Use it constantly, use it now in faith, and you shall surely prevail. But perhaps you have a doubt about interceding for someone who has fallen far into sin.

– quote taken from Intercessory Prayer, a sermon delivered by Charles Spurgeon, British theologian and preacher. Excepted from the book Spurgeon on Prayer. I will grant that the language is flowery, designed to be spoken from the pulpit in the 1800s; nonetheless, I have no more problem with Spurgeon saying "have a doubt about" than I do with him starting the sentence with a preposition.

Analogously, Cicero does not have a doubt about the immortality of the soul, since nature itself has placed in us this conviction. In fact, everyone is concerned with life after death. This is, for Cicero, the surest argument in favor of immortality, even if he does not hesitate to support it with traditional Platonic proofs.

– quote taken from A History of Ancient Philosophy III: Systems of the Hellenistic Age, by Giovanni Reale. Reale was an Italian philosopher who published his five-volume tome in 1975. Blame for this use of the English phrase "have a doubt about" may need to be placed on the shoulders of the translator, John R. Catan; however, given Catan's rather impressive resume, I'm not inclined to categorize this as a rookie mistake.

Now, back to the original question:

To my (British) ear, it never sounds correct to say you have a doubt about something. [emphasis added]

Never is a strong word. I've offered five quotes ranging from the 19th century to the 21st century, ranging from theologians to atheists, ranging in context from philosophy to fiction, from respected authors with roots on either side of the Atlantic. None of them caused me to bristle as though I just heard fingernails scraping on a chalkboard, and none of them made me immediately question if the author was a speaker of Indian English.

Never is a word that spurs pedantic analysis. I usually try to avoid it. In this case, I think there are times when have a doubt about something sounds acceptable, and supporting examples can be found if you're willing to do the research.