Perhaps most commonly associated with the French Foreign Legion, thanks to countless books and movies depicting the Legionnaires in North Africa wearing the iconic white cap cover with neck flap, the havelock was known to be in use since about the middle of the 19th century by the British army in India. Being an extremely useful piece of kit, it's use quickly spread to other armies in other hot regions of the world.
It is sometimes also referred to as a neck curtain, though the official definition says it is a cloth cover for a kepi or military service cap with a flap to protect the back of the neck. Some attribute its origins and name to Sir Henry Havelock, a British officer who served in India, among other places, in the first half of the 19th century. Others say it was actually Sir Henry Hardinage who ordered white cap covers for tropical use in the 1840s prior to leaving for India as Governor-General, the flap being added sometime after. Just as likely, it was an adopted practice after being observed in use in some form by the local populations.
Whatever the case may be, it is a relatively simple item to reproduce.
Basically, whatever hat or cap you want the havelock to sit on top of, measure the circumference, height and diameter of the crown and cut out a circle and a strip out of a sturdy off white cotton fabric, like canvas or twill, that are slightly bigger than the intended cap, and don't forget to add seam allowance.
Sew the two short ends of the strip together, right sides together. Then sew the circle to the strip, again, right sides together.
And here it is flipped right side out and tried on the cap for fit.
The flap part can be just a simple rectangle of fabric with square edges, here it is slightly curved on the bottom edge and a bit narrower at the top. Comes down to individual preference.
It needs to be long enough to cover the back of the neck and wide enough to at least just cover the tips of the ears.
Turn under the seam allowance to finish the edges and sew to the cap part.
And this is what it should look like, more or less, when finished.
This is an interpretation of the Bengal Mounted Police. As there is no known photograph of their uniform we can only speculate as to what they wore.
An interesting aside is that Francis Dickens, son of famous author Charles Dickens, served with the unit before being transferred over to the North West Mounted Police and ended up participating in the suppression of the Riel Rebellion.
Many of the Colonial police forces of the British Empire had a similar look, or similar uniform, usually some rendition of dark blue wool tunic with tan coloured trousers. Below, as an example, is a photo of John Kirkup who served in the Victoria City Police, Mounted Unit, wearing a typical uniform.