Watches with a cover/lid front and back are commonly known as "Hunters" and it is unusual for a Hunter to have anything but an Enamel dial. I have checked our easy reference and am unable to identify the name "Rigorosa" or "Initiative" but the description of the engraving on the case would indicate Swiss manufacture.Your description does not mention any engraving on the movement itself, does this mean there is none? Most quality movements would have at least the makers name engraved. Many pocket watches are "marriages" that is the movement and case are made by different people. The two parts being put together either by a retailer or distributor. So information about the movement is important.
There was no Exposition in London in 1906, however I believe the medals refer to "Trade Prizes". Many different trades would hold annual exhibitions, the sole purpose of which was to pat each other on the back and award medals to each other for their work (very much like todays Oscars for the film industry), 4 medals would indicate that the watch had won 4 different awards possibly in different years. These awards were used by manufacturers in advertising the quality of the watches.
I would like to help identify this item if I can but would need further information and photographs if possible. For instance you don't mention if the watch is key wind or has a button wind, also position of the 12 o'clock in relation to the button is important.