According to the Royal Engineers website, red was chosen for British soldiers based on the national colors of the British flag. Additionally, red dye was also one of the cheapest dyes to produce in the 1500s, which was around the time British soldiers started wearing red.
The first recorded time that British soldiers wore red was with the Yeoman of the Guard during the reign of King Henry VIII. This color remained when the first permanent British army was raised in 1645, and was complemented with white crossbelts and shiny brass.
Although pictures of British soldiers show them wearing red uniforms, historians believe many of the red uniforms faded to a pinkish-brown color over time. This was due to the fact that dyes used for clothing throughout the 1500s and 1700s faded quickly. As soldiers spent time in the rain and sun, their uniforms faded. This meant uniforms had to be replaced or re-dyed often.
Wearing red made British soldiers stand out on the battlefield, but early battle tactics were much different than the tactics that followed the mid-1800s. Until then, camouflage was not something that many countries worried about. This played a major role in America winning its independence from Britain during the Revolutionary War.