The scene in Sharpe is not especially accurate for the period, on several counts. By the point the duel is taking place, which I would assume is circa the Peninsular Campaign, although its been a long time since I watched the series, there were several conventions that would be expected here, and although what is shown on screen reflects a fairly popular image of the duel, wouldn't likely be how a duel would be fought then, although I'll touch briefly on exceptions

In way of brief preamble, the rise of the pistol duel, and the development of the conventions surrounding it during the late 18th century, were intended to place the two parties on as equal a footing as possible and remove skill insofar as possible as a determining factor.

The main way we see this realized is the regulation of shooting to a small window of time, on a signal, so as to minimize the ability to aim. For an English duel of the period, most common would be that both duelists would stand with their pistols pointed up, and on the command "FIRE!" they would have three seconds to do so, counted off by one of the Seconds, after which the command "Stop! would be given and if they had not fired, they had lost their chance.

Firing in turn was not unheard of, to be sure, but it was also something found more commonly in the 18th century than the 19th, so would be a somewhat strange method to conduct the duel seen here, but in those cases, there would usually be some agreement on how long a window was available to fire, usually a few seconds counted off. Likewise it would be strange, but not unheard of to have no count off even in a case of simultaneous fire.

But if we assume that the Seconds in this duel - whose duty it would be to make the arrangements and set the groundrules - were particularly incompetent and allowed for firing in turn, and without regulation of time. In a situation like that, as far as the technically proper conduct goes, there isn't too much to be said. Duels were illegal, and while codes existed, they were guidelines, not hard and fast rules, which reflected conventions of the time. A duel conducted in contravention of those conventions was just as illegal as one in compliance, and the most important factor was the duelists obeying whatever had been agreed to by the Seconds. Prosecutions for dueling were exceedingly rare, and 'following the rules laid down' was a defense which would work in court in the rare case it got that far.

But there was also social opinion of course. Perhaps the best example here would be that of Andrew Jackson who shot and killed Dickinson in a duel in 1807, which while American, does offer us some insight. Firing was simultaneous, but without a time limit, and Jackson decided to hold off firing and let his opponent do so quickly. He was shot, but not mortally so, and in response took very careful, deliberate aim before firing (technically he did it twice. The gun didn't go off so he recocked it and reaimed). Jackson had a burning hatred for Dickinson and is said to have remarked afterwards "If he had shot me through the brain, sir, I should still have killed him”. No legal issues ever came from the duel, but he did endure a good deal of social opprobrium that he had murdered his opponent in cold blood, obeying the letter of the duel, but not the spirit.

We can also find examples to look to in cases where one duelist deloped, that is to say, intentionally missed. This is expanded on more here, but to focus specifically for our case here, while not the same thing, there was some similarity in a duel where one duelist deloped, yet the second duelist having missed as well but not purposefully. This was generally taken to be the end of the duel, the delope was an apology of sorts and the other duelist was able to take their shot. This wasn't always the case though with the second duelist insisting on another fire. Reasons varied for this, in some cases being that they felt the need to be shot at which reflected older conceptions of the duel, but especially in cases where it was clear they wanted another chance to plug their opponent, while nothing could truly prevent it if they did not yet feel satisfied, it nevertheless was something which could reflect poorly on them in the social reporting later.

For sources, please please consult my bibliography here, and if you have any follow-up questions please let me know.