This was the case for at least several hundred years, from the American Revolution up to Victorian times, and the Red Uniform was in use throughout the Empire. By the time of World War I the Red Coat was no longer prevalent.
I'm aware that some units of the British army still wear red when undertaking ceremonial duty, however, it is never worn in the field anymore as far as I can tell. Is it simply down to practical issues? I.e. trying not to stand out?
So, a couple things here: the red coats and the standardization of the military, which are not necessarily exclusive to each other.
With the widespread use of black powder on the battlefield it became important to avoid friendly fire and part of this was to wear easily identifiable colors. The British opted for madder red for their regular infantry as it was a cheap and recognizable color. Officers, who paid for their own uniforms, wore scarlet red derived from the cochineal beetle and was very expensive (at by the time of the American War of Independence.)
This practice was phased out in the Victorian era for a couple different reasons. In hotter weather climates, like Africa, heavy wool coats were not very desirable and the British switched to khaki hot weather uniforms (after being exposed to the fabric in India) and this trend continued into WWI with the British actually being among the very first to adopt drab uniforms suited to the trenches of Western Europe.
Now, the standardization of the military is slightly different. Keep in mind that not everyone in the British army wore red. For example, the Royal Artillery wore blue coats with red facings in a style very similar to what people perceive American infantry of the time. In fact, both British and American artillery wore almost identical uniforms and some American infantry regiments were wearing red coast as well! What we know is that each regiment, on both sides, were prescribed a color combination of coat and facings/cuff colors with the British also using regimental lace around the buttons. The practice of regimental lace is still prominent going into the Zulu wars but not after their uniform reform when it is replaced by shoulder insignia.
Finally, red coats are not exclusive to the British in the Old World either. Both the Spanish and the Portuguese would outfit their regular infantry with red coats, and some Frenchmen would be seen fighting in their small clothes, which included a solid red sleeved vest (basically a coat) in North America against the British so now one must take into account a cockade. A cockade was worn in your hat and the combination of the cockade and coat would identify you, at a glance, with your nation. The British, at least in the 1750s wore black, the French wore white and the Americans (in the 1770s) were wearing a combination black/white.
So, long story short, red coats were worn widely from the early to mid-1600s (sources vary) to about the 1880s.