Why and when did the British Empire decide that their infantry would wear red?

February of 1645 with the passing of the New Model Army ordinance. Red had been popular before that too but this is, to my knowledge, the point when all soldiers, regardless of rank, were ordered to wear the venetia red. There is not rationale provided for the decision but previous documentation (1594) states that coats "be of such colours as you can best provide". Red was fairly easy to produce from the vegetable colours in use at the time and generally required only a single dying to take. It was also vivid and easy to distinguish on the frequently chaotic fields of battle.

Why the French (and Prussians) blue?

The prussians largely adopted the uniform of Brandenburg. Interestingly, this is where the phrase "blue on blue" comes from, it was coined during Moltkes wargames.

The French adopted it from the blue banners already strongly asociated with the Franconia of Charlemange, they fought under blue banners as early as the norman conquest.

Was it an economic decision due to dye availability or something else?

See 1st paragraph.

Where did they get the dyes from?

This was not standarized. Every uniform maker was free to obtain their dyes from wherever but generally vegetable colours in some form where cheapest and most widely used.

Source:

W.Y. Carman - British Military Uniforms From Contemporary Pictures

The Oxford Guide to Heraldry - Woodcock & Robinson