To explain a bit better, was there planning involved in the English military wearing red, the French military wearing Blue, the Russian military wearing green etc, or was it coincidental that they all chose different colours?
Out of the options you've given here it's coincidental. A more accurate answer is - they didn't all wear different colours.
The UK's uniforms were, on the whole (apart from Light Cavalry and Riflemen) red, but so were the Danish uniforms of the time.
The Russians, also on the whole, wore green, but so did soldiers from the Kingdom of Italy, who were in service to Napoleon. French dragoons also wore green coats. Even Napoleon himself preferred to wear the green undress uniform of his Chasseurs à Cheval de la Garde.
White and blue were worn by numerous armies. White was worn by the Spanish, the Saxons and most famously by the Austrians, whilst Prussians, Polish, Swedish and French wore blue.
Lets not even get into how many different colours Hussar regiments wore throughout every country.
But from the amount of colours worn, it's pretty simple to say that there was no planning between countries to lessen confusion on the battlefield, two armies wearing the same colours could very easily meet each other in battle.
This gets even more confusing if greatcoats are added to the mix in winter or particularly bad weather, and entire armies of men wearing grey end up on the battlefield.
It's also worth adding that the uniforms wouldn't stay the proper colour for long either, as natural dyes faded British troops would end up in pink, French uniforms would turn a bluish gray, and Russians would be in an unpleasant yellowed green.
As for the coincidence side of things, the choice of dyes was usually related to policy rather than practicality of a particular colour. The English wore red because it was cheapest for them when their uniforms were first dyed a single colour in the 17th century.
I can't give the reasons for other armies colours, as I don't know them, but the French imported indigo dye for their uniforms, and in 1806 some regiments had to wear white when a British blockade meant they had drastically lower stocks of indigo.
So yeah, tl;dr: Neither, nations equipped their armies with whatever colours they liked, but it was coincidence that those three nations wore different colours.
Please look at this question and the answers.