It's often said that the British Army picked red as the colour of it's uniforms because that was the cheapest dye. Is this true and did similar cost concerns influence other European armies' uniform choices?

Reasons of economy may very well have influenced the adoption of red as the color of soldiers' coats. However, there was a lengthy history of red clothing in English armies stretching back to over a century before its use by Cromwell's New Model Army. Red and blue were the English Royal Livery colors- and subsequently, red coats faced blue were worn by all royal regiments in the British Army. The yeomen of the Guard wore red livery since the 1530's. Red and blue doublets were both popular in the army of Queen Elizabeth I. Furthermore, the symbolic association of red with England is obvious- the Red Cross of St. George served as a national emblem since the fourteenth century.^1

Such symbolic associations informed the adoption of various colors by other national armies. French, Austrian and Spanish soldiers wore white, Russia green, Bavaria light blue, and Prussia dark blue. The color of uniforms could be traced back to some national color, typically the livery of the monarch. Spain and France, both under Bourbon monarchs in the 18th century, wore uniforms in a color associated with their royal family. By regulating the color of its armies clothing, the state reinforced the ties of military force to the ruling power, emphasizing national instead of mercenary ties. Economy entered into the equation, too. As H.W. Koch explains of Prussia's use of dark blue: "Frederick William I was determined to use as much as he could of Prussia's own resources; Prussia produced its own cloth and the only dye which they could manufacture and which could stand all weathers was dark blue."

For further reading, see:

Mollo, John. Military Fashion.

Barthorp, Michael British Infantry Uniforms Since 1660 and British Cavalry Uniforms since 1660.

^1 White, the other prominent color on the flag, was also a popular color for soldiers' uniforms in the reign of Henry VIII. For instance, the London Train Bands were reviewed by the King in the 1539 clad head to toe in white.